13ｨｪｨ 0002ｨｪｨ 00ｨｦ0602ｨｮ ｨｰｨ ｨｲ07 ﾂｨ 06ｨｦ02, ISSN ﾁ

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2 13ｨｪｨ 0002ｨｪｨ 00ｨｦ0602ｨｮ ｨｰｨ ｨｲ07 ﾂｨ 06ｨｦ02, ISSN ﾁ ﾁ ﾁ 罟ｬ ﾁ1 7" ｨｦ : ﾁ ﾁ ｨｪ ﾁ09, , ﾁ ｨｦ ﾂ.ｨｪ. ｨｰ ｨ.ｨｦ ﾁ ﾂ. (0308 ﾁ ﾁ ) ﾁ ｨ. ﾂ ﾁ ｨ ﾁ ｨｮ.ｨｦ ﾁ ｨｦ.08. ｨｮ ｨ.ｨｦ (73), c ｨｪ ﾁ ", ﾁ , ﾁ ﾁ , ｨｪ ﾁ09, ｨｪ ﾁ ", ｨｲ ﾁ ﾁ ｨｰ ｨｰ03, ﾁ , : ｨｮ ﾁ. ｨｹ : 0109 ﾁ ﾁ 01.ｨ., ｨｦ.ｨ., ｨｦ ﾁ ｨ ﾁ ﾁ0007, 05. ｨｪ ﾁ09, ﾁ , ﾁ

3 13ｨｪ ﾁ ﾁ ﾁ (73), 0502 ﾁ { ｨｮ ｨ. ｨｰ. ｨｰ ﾁ ｨｪ ﾁ ｨｲ. ｨｮ09 ﾁ ﾁ ｨ. ｨ. ｨｹ ﾁ ｨｪ ﾁ ｨｮ ﾁ ﾂ. ﾂ. ｨｦ ﾁ0003 ﾁ ﾁ ﾁ ﾁ09, ｨｮ. ｨ ﾁ. ｨｰ ﾁ ｨｮ ﾁ : ｨｬ ﾁ ? 56 ｨｦ ﾁ ｨｦ ﾁ ｨｦ. 05. ｨ ﾁ ﾁ03: ｨｮ 縺 ﾁ ﾁ ﾁ ﾁ ｨｰ { ﾁ ｨｦ. 05. ｨ ﾁ09 { ｨｦ ﾁ ｨｲ " 77

4 ｨｪ ﾁ ｨ. ｨｰ. ｨｰ ﾁ ﾁ , ﾁ. ｨ ﾁ ﾁ ﾁ ( ), ﾁ , , ﾁ ﾁ , ﾁ ｨｦ ﾁ ﾁ ﾂ , ﾁ ﾁ ﾁ ﾁ ﾁ ｨｮ ｨｪ n ﾁ x ﾁ ﾁ ﾁ p(x) = a n x n + a n6ﾓ11 x n6ﾓ a 1 x + a 0 ; (1) ﾁ a n ; a n6ﾓ11 ; : : : ; a 1 ; a , a n ﾙ 0, ﾁ ﾁ ; n a n ﾁ ; a 0 07 ﾁ ; n , deg p(x) = n; a n x n , p(x) = 6x 5 6ﾓ15x 3 +1;2x 2 +x6ﾓ ( ) x a 5 = 6, a 4 = 0, a 3 = 6ﾓ15, a 2 = 1, a 0 = 6ﾓ ﾁ ﾁ ﾁ , , ﾁ ﾁ ｨｲ ﾁ (1) ﾁ ﾁ ﾂ ﾁ , ﾁ ﾁ ｨｦ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ : p(x) = a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n (2) ｨｹ (1) 07 (2) ﾁ ﾁ ﾁ : ﾁ ﾁ ﾁ , , p(x) = 1 + 2x 6ﾓ1 7x 3 + x 4. ｨｲ ﾁ ﾁ ﾁ03 ﾁ ﾁ p(2) = ﾓ = 6ﾓ ﾓ ﾁ ﾁ ｨｦ0007, , (2) p(1) = a 0 + a a n6ﾓ11 + a n 0609 ﾁ ﾁ ﾁ. 01 ﾁ p(x) 07 q(x) ﾁ ﾁ ﾁ ( ﾁ ﾁ ), ﾁ ﾁ02, ﾁ ｨｪ ﾁ02 2

5 13ｨｪ ﾁ ﾁ ﾁ ﾁ03 02 ﾁ ﾁ ﾔ: p(x) ﾔ q(x) , p(x) = 1 + 2x 6ﾓ1 7x 3 + x 4 ﾁ ﾁ (2) = 6ﾓ q(x) = 6ﾓ x ﾁ , q(2) = 6ﾓ = 6ﾓ ﾁ02, 0203 ﾁ p(0) = ﾓ = 1, 09 q(0) = 6ﾓ = 6ﾓ145, p(x) = 1+2x6ﾓ17x 3 +x 4 07 q(x) = 6ﾓ145+5x ﾁ ﾁ0202. ｨｪ p(x) 6ﾘ7 q(x) ﾁ p(x) 07 q(x) ﾁ , ) ﾁ ) ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ : ﾁ ﾁ , 0803 ﾁ ﾁ ( ) ﾁ p(x) ﾔ q(x) 07 p(x) = a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n ; a n ﾙ 0; q(x) = b 0 +b 1 x + +b m6ﾓ11 x m6ﾓ11 +b m x m ; b m ﾙ 0; , n < m. ｨｲ ﾁ03 a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n ﾔ b 0 + b 1 x + + b m6ﾓ11 x m6ﾓ11 + b m x m ﾁ x = 0 ( ﾁ03 ﾁ ) a 0 + a a n6ﾓ11 0 n6ﾓ11 + a n 0 n = b 0 + b b m6ﾓ11 0 m6ﾓ11 + b m 0 m : a 0 = b ﾁ ﾁ03 (6ﾓ5) ﾁ0702 a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n ﾔ b 1 x + + b m6ﾓ11 x m6ﾓ11 + b m x m : (6ﾓ5) a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n ﾔ b 1 x + + b m6ﾓ11 x m6ﾓ11 + b m x m 07, ﾁ ﾁ , a a n6ﾓ11 x n6ﾓ12 + a n x n6ﾓ11 ﾔ b b m6ﾓ11 x m6ﾓ12 + b m x m6ﾓ11 : (6ﾓ56ﾓ5) ｨｦ03 (6ﾓ56ﾓ5) , a 1 = b , ﾁ ﾁ ﾁ ﾁ09, 0703 (6ﾓ5) a 2 = b 2 ; a 3 = b 3 ; a 4 = b 4 ; : : : ; a m = b m : a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n ﾔ a 0 + a 1 x + + a n6ﾓ11 x n6ﾓ11 + a n x n + b n+1 x n b m x m ; ﾔ b n+1 x n b m x m : 0 ﾔ x n+1 (b n+1 + +b m x m6ﾓ1n6ﾓ11 ) 6ﾍ2 0 ﾔ b n+1 +b n+2 x +b m x m6ﾓ1n6ﾓ11 6ﾍ2 b n+1 = b n+2 = = b m = 0 (?!) ﾁ ｨｲ , ﾁ ﾁ03 n < m 0203 ﾁ ｨ , 0203 ﾁ n > m. ｨｮ ﾁ , n = m, ﾁ ﾁ ﾁ 0609 ﾁ , ﾁ , 0403 ﾁ n = m, , a 0 = b 0, a 1 = b 1, a 2 = b 2,... ; a n = b n, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0202,

6 134 ｨ. ｨｰ. ｨｰ ﾁ A, B 07 C, ﾁ , p(x) = A(x 2 + x + 1) + (Bx + C)(x + 1) 07 q(x) = x 2 + 3x ﾁ x. ｨｰ ｨｲ p(x) ﾁ ﾁ ﾁ x ﾁ ﾂ ﾁ0708 p(x) ﾔ q(x). ｨｲ , p(x) = (A + B)x 2 + (A + B + C)x + (A + C): (A + B)x 2 + (A + B + C)x + (A + C) ﾔ x 2 + 3x + 2: , ﾁ p(x) 07 q(x) ﾁ x 0609 ﾁ ﾁ A + B = 1; A + B + C = 3; 63 A + C = 2: ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ03, C = ﾁ A = , ﾁ ﾁ B = 1. ｨｦ080909, A = 0, B = 1, C = ﾁ0708 p(x) ﾔ q(x), ﾁ ﾁ ｨｲ ﾁ ﾁ03 p(x) = q(x) ﾁ x = 6ﾓ11; x = 0; x = 1. ﾂ x A(x 2 + x + 1) + (Bx + C)(x + 1) ﾔ x 2 + 3x + 2 6ﾓ11 A((6ﾓ11) 2 + (6ﾓ11) + 1) + (B(6ﾓ11) + C)((6ﾓ11) + 1) = (6ﾓ11) 2 + 3(6ﾓ11) A( ) + (B 0 + C)(0 + 1) = A( ) + (B 1 + C)(1 + 1) = A, B 07 C ﾁ ﾁ : A = 0; A + C = 2; 63 3A + (B + C) 2 = 6: ｨｰ ﾁ A = 0; B = 1; C = , ﾁ ﾁ : 01 ﾁ n ﾝ ﾁ ﾁ , ﾁ ﾁ n ﾁ090107, n ( ) ﾁ n. 2. ｨｮ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ , ﾁ ﾁ, ﾁ ﾁ ｨｮ ﾁ 0203 ﾁ ﾁ , (x 3 + 6x 6ﾓ1 2) + (6ﾓ1x 3 6ﾓ1 2x 2 + x + 7) = 6ﾓ12x 2 + 7x + 5: ﾂ , ﾁ ﾁ , ﾁ ﾁ : (4x 6ﾓ1 2) + (6ﾓ14x + 2) ﾔ 0.

7 13ｨｪ ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ p(x) 07 q(x) ﾁ ﾁ0301: p(x) 6ﾓ1 q(x) = p(x) + (6ﾓ11) q(x): p(x) ﾁ , q(x) ﾁ , ﾁ ﾁ ﾁ 縺 ﾁ ｨｲ 縺 ﾁ ( ﾁ ) ﾁ , ﾁ ﾁ03 ﾁ ﾁ ﾁ, ﾁ ﾁ , p(x) = 3x 4 + x 2 6ﾓ1 6x q(x) = 2x 4 6ﾓ1 3x 3 + 5x 2 6ﾓ1 12x 6ﾓ1 5: ｨｰ ﾁ ﾁ 縺 : 1) 3x 4 + 0x 3 + 1x 2 6ﾓ1 6x x 4 6ﾓ1 3x 3 + 5x 2 6ﾓ1 12x 6ﾓ1 5 5x 4 6ﾓ1 3x 3 + 6x 2 6ﾓ1 18x 6ﾓ1 3 2) 3x 4 + 0x 3 + 1x 2 6ﾓ1 6x + 2 6ﾓ1 2x 4 6ﾓ1 3x 3 + 5x 2 6ﾓ1 12x 6ﾓ1 5 x 4 + 3x 3 6ﾓ1 4x 2 + 6x ( ﾁ ) 02 ﾁ ﾁ, ﾁ, ﾁ , ﾁ : 3) x 5 6ﾓ1 x 4 + x 6ﾓ1 5 + x 4 6ﾓ1 x 2 + x + 6 x 5 6ﾓ1 x 2 + 2x + 1 4) x 5 6ﾓ1 x 4 + x 6ﾓ1 5 6ﾓ1 x 4 6ﾓ1 x 2 + x + 6 x 5 6ﾓ1 2x 4 + x 2 6ﾓ 縺 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , 0609 ﾁ ﾁ ﾁ ﾁ090107, ﾁ , (x 3 6ﾓ1 1)(x 3 + 2x + 1) = x 6 + 2x 4 + x 3 6ﾓ1 x 3 6ﾓ1 2x 6ﾓ1 1 = x 6 + 2x 4 6ﾓ1 2x 6ﾓ1 1: ｨｮ ﾁ ﾁ 0609 ﾁ ﾁ ﾁ 縺 , ﾁ , , , ﾁ , x x x 2 + 2x + 1 ﾁ x 2 6ﾓ1 3x + 2 x x 5 + 0x 4 + 2x 3 + x 2 + 6ﾓ1 3x 5 + 0x 4 + 0x 3 6ﾓ1 6x 2 6ﾓ1 3x 2x 4 + 0x 3 + 0x 2 + 4x + 2 x 6 6ﾓ1 3x 5 + 2x 4 + 2x 3 6ﾓ1 5x 2 + x , , ﾁ

8 136 ｨ. ｨｰ. ｨｰ ﾁ ﾁ p(x) 07 q(x) 02 ﾁ ﾁ ｨｪ p(x) q(x) , ﾁ t(x), x ﾁ ﾁ ﾁ03 p(x) = q(x) t(x). ﾂ t(x) ﾁ p(x) 0209 q(x). ｨｪ q(x) ﾁ p(x), p(x) p(x) q(x), p(x).q(x) , ﾁ ﾁ ﾁ, t(x) 0609 ﾁ : p n (x) = q m (x) t n6ﾓ1m(x), n 6ﾓ1 m ﾝ , p(x) q(x), p(x) q(x) , p(x) = x 3 6ﾓ q(x) = x 2 +x+1, ﾁ ﾁ03 x 3 6ﾓ1 1 = (x 2 + x + 1)(x 6ﾓ1 1) ﾁ x. ｨｮ , p(x) = 5x q(x) = 2x 2 + 1, p(x) q(x). ｨｪ02 ﾁ , ﾁ 0203 ﾁ ﾁ ﾂ ﾁ p(x) n, n ﾝ 0, q(x) = c, c ﾙ ﾁ , , ﾁ , p(x) = 2x 6ﾓ q(x) = 7, p(x).q(x), ﾁ ﾁ ﾁ ﾁ03 2x 6ﾓ1 5 ﾔ 7 (2 7 x 6ﾓ1 5 7), ﾁ x 6ﾓ ｨｰ , p(x) = 6ﾓ15 07 q(x) = 7. ﾂ p(x).q(x), ﾓ15 ﾔ 7 (6ﾓ1 5 7). ｨｮ ﾁ ﾁ , (6ﾓ15) , ﾁ , , (6ﾓ15) , 0609 ﾁ , ! ｨｪ02 ﾁ , n, n ﾝ 0, ﾁ ﾁ ﾁ ﾁ ﾁ p n (x) 07 q m (x), q m (x) ﾁ ﾁ , ﾁ ﾁ t n6ﾓ1m(x) 07 r s (x) , ﾁ x ﾁ ﾁ ﾁ03 p n (x) = q m (x)t n6ﾓ1m(x) + r s (x); (3) n; m; n 6ﾓ1m; s ﾁ ﾁ ﾁ, r s (x) ﾙ 0, 0803 s < m: ｨｪ r s (x) ﾁ p n (x) 0209 q m (x) ﾁ ﾁ , t n6ﾓ1m(x) ﾁ p n (x) 0209 q m (x) r s (x) ﾔ 0, 0803 p n (x).q m (x): ﾁ ! n p n (x) q m (x), t n6ﾓ1m(x) 0609 ﾁ0209 n 6ﾓ1 m ﾝ n m, ﾁ , , r(x) ﾔ p(x) , x x , 0609 ﾁ , , 0609 ﾁ x ﾁ , 0609 ﾁ ｨｰ09 ﾁ ﾁ03 (3) ﾁ ﾁ ﾁ0702 5x + 6 = (2x 2 + 1) 0 + (5x + 6): ﾁ ﾁ ﾁ ﾁ ﾁ

9 13ｨｪ ﾁ p n (x) q m (x) r s (x), p n (x) c q m (x); c ﾙ 0, r s (x) ｨｹ ﾁ0301: p n (x) = (c q m (x)) t n6ﾓ1m(x) c + r s (x) ﾁ , ﾁ ﾁ 縺 ﾁ 縺 ( ), 縺 ﾁ p(x) q(x) ﾁ ﾁ ﾁ ﾁ ﾁ 縺 ｨｰ 縺 p(x) = 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x q(x) = 2x 3 + x ｨｰ I ﾁ x x x 2, ﾁ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x x 2 II ﾁ ﾁ x , ﾁ ｨｲ ﾁ ﾁ x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 縺 ( ﾁ ). ｨｹ ﾁ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x x 5 + 2x 4 + 2x 2 2x ﾁ : 6ﾓ12x 4 6ﾓ1 2x 3 6ﾓ1 3x 2 + x + 1 ﾂ 罍ﾁ I , ﾁ , ﾁ x. ｨｰ ﾁ (6ﾓ12x 4 ) (6ﾓ12x 4 6ﾓ1 2x 3 6ﾓ1 3x 2 + x + 1) x , ﾁ : 6ﾓ1x ﾁ ﾁ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x x 5 + 2x 4 + 2x 2 2x 2 6ﾓ1 x 6ﾓ12x 4 6ﾓ1 2x 3 6ﾓ1 3x 2 + x + 1 III ﾁ , ﾁ03 II , : ﾁ (6ﾓ1x) ﾁ ﾁ (6ﾓ12x 4 6ﾓ1 x 3 6ﾓ1 x) ﾁ , 0609 ﾁ ﾓ12x 4 6ﾓ1 2x 3 6ﾓ13x 2 +x+1. ﾂ 縺 , ﾁ (6ﾓ1x 3 6ﾓ13x 2 +2x+1). ｨｲ ﾁ ﾂ ({0,5): 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x x 5 + 2x 4 + 2x 2 2x 2 6ﾓ1 x 6ﾓ1 0;5 6ﾓ12x 4 6ﾓ1 2x 3 6ﾓ1 3x 2 + x + 1 6ﾓ1 6ﾓ12x 4 6ﾓ1 x 3 6ﾓ1x ﾁ : 6ﾓ1x 3 6ﾓ1 3x 2 + 2x + 1 ｨｪ02 ﾁ , ﾁ ﾁ , q(x). ｨｹ

10 138 ｨ. ｨｰ. ｨｰ ﾁ r(x) p(x) 0209 q(x) ﾁ , ﾁ , ｨｲ09 ﾁ ﾁ ﾁ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x x 5 + 2x 4 + 2x 2 2x 2 6ﾓ1 x 6ﾓ1 0;5 6ﾓ12x 4 6ﾓ1 2x 3 6ﾓ1 3x 2 + x + 1 6ﾓ1 6ﾓ12x 4 6ﾓ1 x 3 6ﾓ1x 6ﾓ1x 3 6ﾓ1 3x 2 + 2x + 1 6ﾓ1 6ﾓ1x 3 6ﾓ10; 5x 2 6ﾓ1 0; 5 6ﾓ12;5x 2 + 2x + 1;5 ｨｲ x 2 6ﾓ1 x 6ﾓ1 0; , r(x) = 6ﾓ12;5x 2 + 2x + 1; p(x) 0209 q(x). ｨｰ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 = (2x 3 + x 2 + 1) (2x 2 6ﾓ1 x 6ﾓ1 0; 5) + (6ﾓ12; 5x 2 + 2x + 1; 5): ｨｦ : 4x 5 6ﾓ1 2x 3 6ﾓ1 x 2 + x + 1 2x 3 + x 2 = 2x ﾓ1 x 6ﾓ1 0; 5 + 6ﾓ12; 5x2 + 2x + 1; 5 2x 3 + x 2 : + 1 ｨ 縺 ﾁ ﾁ 縺 ｨｹ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ. ｨｹ ﾁ , ( ﾁ ﾁ0707) 07 ﾁ ﾁ p(x) = x 3 + x q(x) = x 2 + x 6ﾓ1 2. ｨｦ ﾁ ﾁ , ﾁ ｨｲ ﾁ ﾁ , ﾁ ﾁ : x 3 + x + 4 ﾔ (x 2 + x 6ﾓ1 2)(ax + b) + (cx + d); (4) ax+b cx+d ( ) a; b; c; d, , , ﾁ ﾁ0300 ﾔ, , ﾁ ﾁ03 (4) 05 ﾁ ﾁ0301, ﾁ ﾁ0301, ﾁ x ﾁ a; b; c; d ﾁ ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ : x 3 + x + 4 ﾔ ax 3 + (a + b)x 2 + (6ﾓ12a + b + c)x + (6ﾓ12b + d): ﾁ ﾁ ﾁ , ﾁ ﾁ x ﾁ0207 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , x 3 : 1 = a; x 2 : 0 = a + b; x 1 : 1 = 6ﾓ12a + b + c; x 0 : 4 = 6ﾓ12b + d:

11 13ｨｪ ﾁ ｨｦ : a = 1, b = 6ﾓ11, c = 4, d = : x 3 + x + 4 = (x 2 + x 6ﾓ1 2)(x 6ﾓ1 1) + 4x + 2; x 6ﾓ ; 4x x 3 + x x 2 + x 6ﾓ1 2. ﾂ ﾁ ﾁ ﾁ03 (4) ﾁ x ﾁ , ﾁ ﾁ09, a; b; c; d ﾁ ﾁ x ﾁ , a; b; c; d , ﾁ ﾁ ｨｮ ﾁ , ﾁ x. ｨｲ , x = 1 07 x = 6ﾓ12 { ﾁ x 2 + x 6ﾓ ﾁ x: 0, 1, {2, { ﾁ09 (4) : x = 0: 4 = 2b + d; x = 1: 6 = c + d; x = 6ﾓ12: 6ﾓ16 = 6ﾓ12c + d; x = 6ﾓ11: 2 = 6ﾓ12(6ﾓ1a + b) + (6ﾓ1c + d): ｨｹ ﾁ ﾁ , ﾁ a = 1, b = 6ﾓ11, c = 4, d = ﾁ , ﾁ ﾁ , 0308 ﾁ : ﾁ ﾁ03 ﾁ ﾁ ﾁ ﾁ p(x) 07 q(x) ﾁ ﾁ d(x), ﾁ p(x) 07 q(x) ﾁ ﾁ p(x) 07 q(x) ﾁ D(x), ﾁ p(x) 07 q(x) D(x) = (p(x); q(x)). ｨｦ ﾁ , : 1) , ; 2) ﾁ p(x) 07 q(x) 05 ﾁ ﾁ ﾁ , ( ﾁ ) ﾁ p(x) 07 q(x), ﾁ ﾁ p(x) = x 3 6ﾓ1 4x 2 6ﾓ1 5x 07 q(x) = 2x 4 6ﾓ1 x 3 6ﾓ1 3x 2 : ｨｰ ｨｰ ﾁ , p(x) = x(x + 1)(x 6ﾓ1 5); q(x) = x 2 (x + 1)(2x 6ﾓ1 3): ｨｦ ﾁ , ﾁ p(x) 07 q(x) D(x) = x(x + 1), ﾁ ﾁ, D(x), ﾁ p(x) 07 q(x), , D 1 (x) = 6ﾓ1x(x + 1), D 2 (x) = = 0;5x(x + 1). ﾂ D(x), D 1 (x), D 2 (x) ﾁ , ﾁ ﾁ ,

12 1310 ｨ. ｨｰ. ｨｰ ﾁ ﾁ , , ﾂ D(x) = x(x + 1) p(x) 07 q(x) , ﾁ ﾁ ﾁ. ｨｹ ﾁ ﾁ ﾁ ﾁ ﾁ ｨ ﾁ ﾁ p(x) 07 q(x) ﾁ ﾁ ﾁ ﾁ: p(x) = q(x)t 1 (x) + r 1 (x); q(x) = r 1 (x)t 2 (x) + r 2 (x); r 1 (x) = r 2 (x)t 3 (x) + r 3 (x); r s6ﾓ12 (x) = r s6ﾓ11 (x)t s (x) + r s (x); r s6ﾓ11 (x) = r s (x)t s+1 (x); (p(x); q(x)) = r s (x) ﾁ ﾁ p(x) = x 5 + x 4 + 2x 3 + 3x q(x) = 2x 4 + 3x 3 + x 2 + 2x + 1: ｨｰ ｨｮ p(x) q(x). ｨｰ p(x) 0209 q(x) 縺 , , ﾁ ﾁ ﾁ. ｨｹ , ﾁ , ﾁ ｨｦ080909, ﾁ p(x) : 2x 5 + 2x 4 + 4x 3 + 6x x 4 + 3x 3 + x 2 + 2x + 1 2x 5 + 3x 4 + x 3 + 2x 2 + x x 6ﾓ1x 4 + 3x 3 + 4x 2 6ﾓ1 x , ﾁ , ﾁ ﾓ1x 4 + 3x 3 + 4x 2 6ﾓ1 x : 6ﾓ12x 4 + 6x 3 + 8x 2 6ﾓ1 2x + 8 2x 4 + 3x 3 + x 2 + 2x + 1 6ﾓ12x 4 6ﾓ1 3x 3 6ﾓ1 x 2 6ﾓ1 2x 6ﾓ1 1 6ﾓ11 9x 3 + 9x ｨｪ ﾁ r 1 (x) = 9x 3 + 9x ｨｰ ﾁ ( ﾁ ﾁ r 1 (x)), , ﾁ , q(x) 0209 r 1 (x): 2x 4 + 3x 3 + x 2 + 2x + 1 x 3 + x x 4 + 2x 3 + 2x 2x + 1 x 3 + x x 3 + x ﾁ , , 0609 ﾁ ｨｮ ﾁ , ﾁ , ﾁ ｨｦ (p(x); q(x)) = x 3 + x , ﾁ , ﾁ ｨｲ , ﾁ ﾁ 0609 ﾁ , ｨｮ x2 + 4x 6ﾓ1 5 5x 3 + 6x 6ﾓ1 11.

13 13ｨｪ ﾁ ｨｰ (x 2 + 4x 6ﾓ1 5; 5x 3 + 6x 6ﾓ1 11), ﾁ ﾁ09 ﾁ ﾁ ｨｦ x 3 + 6x 6ﾓ1 11 x 2 + 4x 6ﾓ1 5 5x x 2 6ﾓ1 25x 5x 6ﾓ1 20 6ﾓ120x x 6ﾓ1 11 6ﾓ120x 2 6ﾓ1 80x x 6ﾓ ﾁ x 6ﾓ ｨｰ x 2 + 4x 6ﾓ1 5 x 6ﾓ1 1 x 2 6ﾓ1 x x + 5 5x 6ﾓ1 5 5x 6ﾓ , 0609 ﾁ ﾁ ｨｮ ﾁ , ﾁ , : (x 2 + 4x 6ﾓ1 5; 5x 3 + 6x 6ﾓ1 11) = x 6ﾓ1 1: ｨｰ x 6ﾓ x2 + 4x 6ﾓ1 5 5x 3. ﾂ x 6ﾓ x 2 + 4x 6ﾓ1 5. x = ﾓ ｨｲ , x 2 + 4x 6ﾓ1 5 = (x 6ﾓ1 1)(x + 5) x 3 + 6x 6ﾓ1 11. x = ﾓ ｨｲ , 5x 3 + 6x 6ﾓ1 11 = (x 6ﾓ1 1)(5x 2 + 5x + 11) x2 + 4x 6ﾓ1 5 5x 3 + 6x 6ﾓ1 11 = (x 6ﾓ1 1)(x + 5) (x 6ﾓ1 1)(5x 2 + 5x + 11) = x + 5 5x x ｨｰ ﾁ ｨ ｨｰ ﾁ0707, ﾁ ｨｪ ﾁ ﾁ (ｨｦ ); , ﾁ , ｨｬ ﾁ ﾁ09 ｨｪ ﾁ02 ﾁ ﾁ [email protected] 2 ｨｮ ﾁ ﾁ ﾁ ﾁ x 6ﾓ1 a; ﾁ0201, ﾁ , ﾁ ﾁ 0209 x 6ﾓ1a, ,

14 ｨｲ. ｨｮ09 ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ｨ ﾁ ﾁ ﾁ07 ﾁ ﾁ , , , ﾁ ﾁ , ﾁ ﾁ ﾁ. ｨｮ ﾁ ﾁ ﾁ ﾁ ﾁ ( ﾁ , ﾁ090801, ). ﾂ ﾁ, ﾁ , , , , f(x) = 09 n cos nx+b n sin nx+ + a 0, 0803, ﾁ , , ｨｮ , ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ , ﾁ ﾂ ｨｪ , , , ｨｮ. ｨｬ ﾁ ﾁ ? ( ｨｪ) , ﾁ f(y; z), ﾁ cos x, 09 z 0209 sin x , ﾁ ﾁ ｨｪ cos x (05ｨｪ) sin x (ｨｮｨｪ) ｨｪ: sin(x + =6); sin 2x = 2 sin x cos x. ｨｦ cos(n + 1)x = cos nx cos x 6ﾓ1 sin nx sin x; sin(n + 1)x = sin nx cos x + cos nx sin x , ﾁ n cos nx 07 sin nx 00ｨｪ. ﾂ ﾁ , ｨｮ , ( ) , ﾁ ﾁ ｨｦ, , , (06 + 1)(06 + 2) 07 (06 + 1) , ﾁ ﾁ , , sin 3 x + cos 2 x sin x 07 sin x ｨｪ, ｨｪ cos x, 09 z 0209 sin x ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a: ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ｨｪ ﾁ ﾂ ﾁ , ﾁ ﾁ ｨｪ ﾁ 1 ｨｪ ｨｮ ", ｨｪ ", ｨｪ.: 03ｨ ｨｲｨｦｨｮ, ｨｮ

15 ﾁ , ﾁ ﾁ , , , ﾁ ﾁ03, ﾁ ﾁ. ｨｮ ﾁ ﾁ : 0108 ﾁ ( ﾁ ) n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a ﾔ 0; 0803 a n = b n = = a = ﾁ ( ) ﾁ 00ｨｪ 09 n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a ﾁ a i 07 b i ﾁ (i > 0) i cos ix + b i sin ix ( = (2k=i), k , ) , ﾁ ｨｦ ﾁ n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a; 09 n cos n(x + 00) + b n sin n(x + 00) + 09 n6ﾓ11 cos(n 6ﾓ1 1)(x + 00) + b n6ﾓ11 sin(n 6ﾓ1 1)(x + 00) + + a = = 09 n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a: i cos i(x + 00) + b i sin i(x + 00) 09 ﾁ i cos ix + d i sin ix: 07 i cos ix + d i sin ix ﾔ 09 i (cos ix cos it 6ﾓ1 sin ix sin it) + b i (sin ix cos it + cos ix sin it) ﾔ ﾔ (09 i cos i00 + b i sin i00) cos ix + (b i cos i00 6ﾓ1 a i sin i00) sin ix: , n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a ﾔ ﾔ 07 n cos nx + d n sin nx + c n6ﾓ11 cos(n 6ﾓ1 1)x + d n6ﾓ11 sin(n 6ﾓ1 1)x + + a: ﾁ ﾁ , , ﾁ i ﾁ ﾁ ﾁ03 a i = c i ; b i = d i, , ﾁ n 09 i cos i(x + 00) + b i sin i(x + 00) ﾔ 07 i cos ix + d i sin ix ﾔ 09 i cos ix + b i sin ix; i cos ix + b i sin ix , ﾁ ( ﾁ ). ｨｦ ﾁ , n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a ﾔ 0: 09 n cos nx + b n sin nx ﾔ 6ﾓ1(09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a): ﾁ (2=n) ｨｲ ﾁ , ﾁ ﾁ ﾁ , , , 07 ﾁ , 0609 ﾁ ｨｦ (2=n) i cos ix + b i sin ix , n cos nx + b n sin nx ﾔ a n cos 09 + b n sin 09 ﾔ 09 ( ﾁ ﾁ ﾁ nx). ｨｮ ﾁ ; a n 07 b n ﾁ

16 ｨｲ. ｨｮ09 ﾁ ﾁ : ﾁ ﾁ , ﾁ , , ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ n cos nx+b n sin nx+09 n6ﾓ11 cos(n6ﾓ11)x+b n6ﾓ11 sin(n6ﾓ11)x+ +a, ﾁ ﾁ n6ﾓ11 cos(n6ﾓ11)x+b n6ﾓ11 sin(n6ﾓ11)x+ +a. ｨｹ ﾁ ﾁ ﾁ ｨｦ080909, ﾁ ﾁ ﾁ n = 1, a 1 cos x + b 1 sin x + a, ﾁ , ﾁ a 1 cos x + b 1 sin x ﾁ ﾁ c cos(x + '), = a b 2 1, 09 ' ﾁ ﾁ , ﾁ ﾁ n = k 6ﾓ n = k ﾁ ﾁ ﾁ ﾁ ﾁ ( n 0209 k), ﾁ ﾁ , ﾁ ( n 0209 k) ﾁ ﾁ n = k , ﾁ ﾁ n = k ﾁ ﾁ ﾁ n = k ﾁ ﾁ ﾁ (ｨｮ03) 00ｨｪ, ﾁ ﾁ , n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a; n 07 b n ﾁ ｨｮ ｨｪ, ﾁ ﾁ , ｨｪ. ｨｨ , ﾁ ﾁ ｨｪ ｨｪ f(x) ﾁ g(x; y) , f(x) ﾔ g(cos x; sin x) ﾁ , , , , ｨｪ sin 3 x + cos 3 x, , ｨｪ , ﾁ ﾁ ｨｮ03 00ｨｪ f(x) = 09 n cos nx + b n sin nx + 09 n6ﾓ11 cos(n 6ﾓ1 1)x + b n6ﾓ11 sin(n 6ﾓ1 1)x + + a 1 cos x + b 1 sin x + a: f(x) ﾁ ﾁ , n. ｨｮ ｨｪ, ﾁ ﾁ , , ｨｮ03 00ｨｪ ﾁ ﾁ ( ﾁ0302 0) ﾁ ｨｲ ﾁ ｨｮ03 00ｨｪ cos n x 07 sin n x n ｨｮ ｨｪ cos n x + sin n x n ﾂ ｨｮ03 02 ﾁ ｨｪ ﾁ ? , ? ｨｮ03 00ｨｪ sinkx sin lx sin mx a; b; c; d; e k; l; m 0609 ﾁ ﾁ03 05 ﾁ ﾁ0301? a sin x + b sin 2x + c sin 3x + d sin 4x + e sin 5x = 4 sin kx sin lx sin mx

17 ﾁ f(x) = 2 sin 6x + 5 cos 6x 6ﾓ1 sin 2x 6ﾓ1 3 cos 2x ﾁ f(x) = 2 sin(x=2) + 3 sin(x=3) + 4 cos(x=5) ﾁ f(x) = 2 sin(3x=) + 3 cos(2x=) ﾁ f(x) = 2 cos(x) + 3 sin(2x) + 4 cos x + 5 sin x? ﾁ ﾁ ﾁ n, ﾁ ﾁ ﾁ cos(nx) 07 sin(nx) 0209 cos( n x) 07 sin( n x)? ﾁ f(x) = 2 cos( ﾌ 206) + 3 sin( ﾌ 306) + 5 cos( ﾌ 506)? ﾁ ｨｦ03 ﾁ , f(x), , ﾁ070801, ﾁ , ﾁ ﾁ f(x) + f(6ﾓ1x) f(x) 6ﾓ1 f(6ﾓ1x) ｨｹ ﾁ ﾁ ﾁ ﾁ f(x) (f (x) 07 f (x)). ｨｲ , f(x) 00ｨｪ, ｨｪ. ｨｮ , sin 2 x = 1 6ﾓ1 cos 2 x, ﾁ ｨｪ ﾁ ﾁ ﾁ a cos k x sin l x, l 0609 ﾁ , l = l = 1, ﾁ ｨｪ ﾁ ﾁ f(x) = f 1 (cos x) + sin 06f 2 (cos x), f 1 07 f f 1 (cos x) , 09 sin 06f 2 (cos x) , ｨｪ f 1 (cos x) 07 sin 06f 2 (cos x) ｨ f 1 07 f 2 ( )? ｨｪ , ﾁ ﾁ09 02 ﾁ ﾁ , ﾁ ﾁ cosx , 0803 f , cos x , sin , f ﾂ02 ﾁ0302: ﾁ07 ﾁ f 1 07 f 2 ﾁ , ﾁ ｨｪ ｨｮ032. ｨｮ , ﾁ ﾁ ﾁ03, ﾁ ｨｮ03, ｨｮ ｨｪ f 1 (cos x) + sin 06f 2 (cos 06) ﾁ f 1 (x) 07 06f 2 (x), ﾁ ﾁ ﾁ0202 0, ｨｪ ﾁ ﾁ ﾁ0302 0, ｨｪ ﾁ ﾁ ﾁ ﾁ0302 0, ｨｪ 0609 ﾁ ﾁ ﾂ , 0703 ﾁ ｨｪ, ﾁ ﾁ ﾁ03 ( ) ﾁ ﾁ ｨｪ , ﾁ ｨｪ ﾁ ﾁ ﾁ ｨｮ ｨｪ ﾁ , ﾁ ﾁ f 1 (x) 07 f 2 (x) 02 ﾁ09 00ｨｪ ﾓ ﾁ ﾁ ﾁ n 07 m, n ﾝ m. ｨｹ , f 1 (x) = (a n cos n x+ +a)+sin x(b n6ﾓ11 cos n6ﾓ11 x+ +b), a n 07 b n6ﾓ ﾁ ( n = 0, 0803 a n ﾁ0203 0); f 2 (x) = (c m cos m x + + c) + sin x(d m6ﾓ11 cos m6ﾓ11 x + + d) n = m f 1 (x) + f 2 (x) = ((a n + c n ) cos n x + ) + sin x((b n6ﾓ11 + d n6ﾓ11 ) cos n6ﾓ11 x + ); , , ﾁ n. ｨ n > m f 1 (x) + f 2 (x) = (a n cos n x + ) + sin x(b n6ﾓ11 cos n6ﾓ11 x + );

18 ｨｲ. ｨｮ09 ﾁ ﾁ ﾁ ﾁ ﾁ , n, 09 ﾁ03 ﾁ n6ﾓ1 1. ｨ a n 07 b n6ﾓ , ﾁ0209 n f 1 (x) 07 f 2 (x) 02 ﾁ09 00ｨｪ ﾁ ﾁ ﾁ n 07 m, n ﾝ m. ｨｹ , f 1 (x) = 09 n cos nx + b n sin nx + + a 1 cos x + b 1 sin x + a; n 07 b n ﾁ0203 0, n = m f 2 (x) = c m cos mx + d m sin mx + + c: f 1 (x) + f 2 (x) = ((a n + c n ) cos nx + (b n + d n ) sin nx + ); , , ﾁ n. ｨ n > m ﾁ , f 1 (x) + f 2 (x) n cos nx 07 b n sin nx. ｨｲ ﾁ0209 n ﾁ ｨｪ , ﾁ ﾁ f 1 (x) 07 f 2 (x) 02 ﾁ09 00ｨｪ ﾁ ﾁ ﾁ n 07 m. ｨｹ , f 1 (x) = (a n cos n x + + a) + sin x(b n6ﾓ11 cos n6ﾓ11 x + + b) f 2 (x) = (c m cos m x + + c) + sin x(d m6ﾓ11 cos m6ﾓ11 x + + d); a n 07 b n6ﾓ c m 07 d m6ﾓ ﾁ f 1 (x)f 2 (x) = ((a n cos n x + : : : ) ﾁ ﾁ (c m cos m x + + c) + (1 6ﾓ1 cos 2 x)(b n6ﾓ11 cos n6ﾓ11 x + + b)(d m6ﾓ11 cos m6ﾓ11 x + + d)) = = (a n c m cos n+m x 6ﾓ1 b n6ﾓ11 d m6ﾓ11 cos n+m x + ) + sin x(b n6ﾓ11 c m cos m+n6ﾓ11 x + + a n d m6ﾓ11 cos m+n6ﾓ11 x + ): , a n c m b n6ﾓ11 d m6ﾓ11 07 b n6ﾓ11 c m + a n d m6ﾓ (a n c m b n6ﾓ11 d m6ﾓ11 ) 2 + (b n6ﾓ11 c m + a n d m6ﾓ11 ) 2 = = a 2 nc 2 m 6ﾓ1 2a n c m b n6ﾓ11 d m6ﾓ11 + b 2 n6ﾓ11d 2 m6ﾓ11 + b 2 n6ﾓ11c 2 m + 2b n6ﾓ11 c m a n d m6ﾓ11 + a 2 nd 2 m6ﾓ11 = = a 2 nc 2 m + b 2 n6ﾓ11d 2 m6ﾓ11 + b 2 n6ﾓ11c 2 m + a 2 nd 2 m6ﾓ11 = (a 2 n + b 2 n6ﾓ11)(c 2 m + d 2 m6ﾓ11): , ﾁ ｨ a n c m b n6ﾓ11 d m6ﾓ11 07 b n6ﾓ11 c m + + a n d m6ﾓ ﾁ0202 0, (a n c m b n6ﾓ11 d m6ﾓ11 ) 2 + (b n6ﾓ11 c m + a n d m6ﾓ11 ) ﾁ f 1 (x) 07 f 2 (x) 02 ﾁ09 00ｨｪ ﾁ ﾁ ﾁ n 07 m. ｨｹ , f 1 (x) = 09 n cos nx + b n sin nx + + a 1 cos x + b 1 sin x + a; f 2 (x) = c m cos mx + d m sin mx + + c; a 2 n + b 2 n6ﾓ11 ﾙ 0 07 c2 m + d 2 m6ﾓ11 ﾙ 0; ﾁ ﾁ a k cos kx 07 b k sin kx 07 k < n, 09 ﾁ ﾁ c l cos lx 07 d l sin lx 07 l < m. ｨｲ f 1 (x)f 2 (x) = (09 n cos nx + b n sin nx)(c m cos mx + d m sin mx) + ; ﾁ ﾁ a k c l cos kx cos lx; a k d l cos kx sin lx; b k d l sin kx sin lx , k 07 l ﾁ ﾁ ﾁ n 07 m

19 ﾁ ﾁ n m, k + l < n + m ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ a cos ix 07 b sin ix , I < n+m, n+m , ﾁ0209 n + m ﾁ : (09 n cos nx + b n sin nx)(c m cos mx + d m sin mx) (09 n cos nx + b n sin nx)(c m cos mx + d m sin mx) = = a n c m cos nx cos mx + b n c m sin nx cos mx + 09 n d m cos nx sin mx + b n d m sin nx sin mx = = (1=2)a n c m (cos(n + m)x + cos(n 6ﾓ1 m)x) + (1=2)b n d m (6ﾓ1 cos(n + m)x + + cos(n 6ﾓ1 m)x) + (1=2)b n c m (sin(n + m)x + sin(n 6ﾓ1 m)x) + (sin(n + m)x + sin(n 6ﾓ1 m)x) = = (1=2)(a n c m 6ﾓ1 b n d m ) cos(n + m)x + + (1=2)(b n c m + 09 n d m ) sin(n + m)x + (1=2)(a n c m + b n d m ) cos(n 6ﾓ1 m)x + + (b n c m 6ﾓ1 09 n d m ) sin(n 6ﾓ1 m)x: ﾂ a n c m + b n d m 07 b n c m 6ﾓ1 09 n d m a n c m b n6ﾓ11 d m6ﾓ11 07 b n6ﾓ11 c m + a n d m6ﾓ ﾁ09, ﾁ ﾁ ﾁ ﾁ09 (ac 6ﾓ1 bd) 2 + (bc + ad) 2 = (a 2 + b 2 )(c 2 + d 2 ): ｨｲ , ﾁ ﾁ ﾁ ﾁ ｨｪ, ﾁ ( ). ｨｹ , f 1 (x) + f 2 (x) + f 3 (x) , (f 1 (x) + f 2 (x)) + f 3 (x) , ﾁ , ﾁ , ﾁ ﾁ ﾁ cos n x sin m x. ｨｲ , = cos x = sin , ﾁ ｨ ｨｪ 09 = cos 0 x ﾔ = sin 0 06 ﾔ ﾁ ﾁ , n > 1 ﾁ a n ﾁ n , 0609 ﾁ , ﾁ ｨｪ ﾁ a cos n x + b cos n6ﾓ11 x sin x a = b = 0, , a 07 b , ｨｪ ﾁ0209 n. ｨ : a cos n x + b cos n6ﾓ11 x sin x = cos n6ﾓ11 x(a cos x + b sin x) ﾁ ﾁ ｨｪ, ﾁ0202 n 6ﾓ , , ｨｪ , ﾁ ﾁ , ｨｪ f(x) , 0803 f(x) ﾔ ﾁ0209 0, 0803 f(x) ﾔ 09, ﾙ 0. ｨｮ ﾁ ﾁ0209 n, n > f(x) ﾁ ｨｪ ﾁ a n cos n x + b n6ﾓ11 cos n6ﾓ11 x sin x, a 2 n+b 2 n6ﾓ11 ﾙ ｨｪ ｨｪ a k cos k x+ b k cos k6ﾓ11 x sin x , k < n 07 a 2 k + b2 k6ﾓ11 ﾙ ﾁ ﾁ0209 k. ｨｦ, ﾁ ｨｪ, , ﾁ0209 n , ｨｪ f(cos x) 07 f(sin x) 0703 ﾁ f sin cos x , ﾁ ﾁ f(cos x; sin x) = , ﾁ (cos x; sin x) , , ｨｮ ﾁ ﾁ ﾁ ﾁ ﾁ

20 ｨｲ. ｨｮ09 ﾁ ﾁ09 ｨ , , n , n , ﾁ f(cos x) = 0 07 f(sin x) = 0, f(x) n , n ﾁ [0; ] 07 [6ﾓ1=2; =2] ﾁ ﾁ ﾁ f(cos x) = n ﾁ [6ﾓ1; ], 09 f(sin x) = 0 ﾁ [6ﾓ1=2; (3)=2]. ｨ ﾁ ﾁ f(cos x; sin x) = 0, f(y; z) ﾁ ? ﾁ ﾁ ﾁ ﾁ g(cos x) = 0, g(y) y = f(x) , , y = f(x)f(6ﾓ1x) ﾁ f(x) f ﾂ , f ﾁ ﾁ f (x) = 0 ﾁ ﾁ , ﾓ ﾁ f(x) = , ｨｪ ﾁ ｨｪ, ﾁ , ｨｮ ﾁ0702 g(cos x), g(y) ﾁ ﾁ , f , g, 0609 ﾁ0209 2n. ｨｮ ﾁ , ｨｪ n n ﾁ [6ﾓ1; ], cos x ｨｪ n ﾁ , ﾁ , n ﾁ 02 ﾁ ﾁ ｨｪ n n ﾁ ] 6ﾓ1 ; ] ﾁ , , f(x) = f 1 (cos x) + sin 06f 2 (cos x) f(6ﾓ1x) = = f 1 (cos x) 6ﾓ1 sin 06f 2 (cos x); f (x) = f(x)f(6ﾓ1x) = (f 1 (cos x) + sin 06f 2 (cos x))(f 1 (cos x) 6ﾓ1 sin 06f 2 (cos x)) = = f 2 1 (cos x) 6ﾓ1 f 2 2 (cos x)(1 6ﾓ1 cos 2 x): f(06) ﾁ [0; ] k , , ﾓ ﾁ ﾁ f 1 (cos 09) + sin 09f 2 (cos 09) = 0 07 f 1 (cos 09) + sin 09f 2 (cos 09) = f 1 (cos 09) = sin 09f 2 (cos 09) = 0, sin ﾁ0302 0, 0803 f 1 (cos 09) = f 2 (cos 09) = f (x) = g(cos x) = f 2 1 (cos x) 6ﾓ1 f 2 2 (cos x)(1 6ﾓ1 cos 2 x), cos g(y) , n 6ﾓ12k. ｨｲ , 09 f (x) ﾁ [0; ] , n 6ﾓ1 2k b, b 0609 ﾁ ﾁ b 07 6ﾓ1b f(06).ｨｲ , f(06) ﾁ ] 6ﾓ1 ; ] , f (x) ﾁ [0; ]. ｨｲ ﾁ f(06) ﾁ ] 6ﾓ1 ; ] , k + (2n 6ﾓ1 2k) = 2n ﾁ a; b; c; d TM y = a cos 3 x + b cos 2 x sin x + c cos x sin 2 x + d sin 3 x 05 ﾁ ? ? ﾂ , ﾁ ﾁ ﾁ ﾁ ｨｪ: ﾁ f(y; z) , f(cos x; sin x) ｨｪ ﾁ ( , ｨｪ ﾁ ﾁ0302 0, , , , ｨｦ ｨｪ: 1; cosx; cos 2 x; cos 2 x + sin 2 x; cos 3 x + sin 3 x f(y; z) n ﾁ ｨｪ ｨｪ f(cos x; sin x) n? 2.4. ｨｪ ｨｪ ? 2.5. ｨｪ ｨｪ ?

21 ｨｮ ｨｪ cos n x, sin n x 07 cos n x + sin n x n ﾂ , ｨｪ cos n x + sin n x ﾁ ﾁ0209 n? ｨ n ? a n (x) = cos n x + sin n x , ﾁ a n+1 (x) a n (x) 07 a n6ﾓ11 (x) : ﾁ , y n+1 + z n+1 = (y n + z n )(y + z) 6ﾓ1 yz(y n6ﾓ11 + z n6ﾓ11 ): ｨｪ cos 2n x + sin 2n x : n ｨｰ ﾁ sin 6 x + cos 6 x = a ｨｪ 3 cos 5 x + sin 5 x 6ﾓ1 sin x. ｨｰ ﾁ cos 5 x + sin 5 x 6ﾓ1 sin x = y = f(x) f(x) = f 1 (x)+f 2 (x), f , 09 f ﾂ , f(x) ﾁ, ? = cos n x + sin m x, n; m m = 2 cos 3x + 3 tg 5x + 5=(sin 7x): ﾁ ｨｪ y = f(x) = cos 3 x + sin 3 x + (1=2) sin 2x 6ﾓ1 1? , ﾁ cos 3 x + sin 3 x + (1=2) sin 2x 6ﾓ1 1 = 0, ﾁ = 09 + ' ﾁ ｨｪ y = f(x) = sin 2x 6ﾓ1 12(sin x 6ﾓ1 cos x) + 12? ｨｰ ﾁ sin 2x 6ﾓ1 12(sin x 6ﾓ1 cos x) + 12 = ﾁ ｨｮ03 00ｨｪ, ﾁ , = ' ﾁ ｨｪ? ﾂ , = (1=4) 05 ﾁ ｨｪ ﾁ ﾁ , ｨｪ ﾁ ﾁ ﾁ g(cos(x 6ﾓ1 1=4)) h(cos x + sin x), g 07 h ? ﾂ , = 6ﾓ1(1=4) 05 ﾁ ｨｪ ﾁ ﾁ , ｨｪ ﾁ ﾁ ﾁ g(cos(x + 1=4)) h(cos x sin x), g 07 h ? ﾁ ﾁ ? ｨｪ , ﾁ ﾁ ﾁ , ﾁ , 0803

22 ｨｲ. ｨｮ09 ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ a k x k , , ﾂ , , ﾁ ｨｪ ﾁ ｨ, ﾁ , ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ , ﾁ09, ( ﾁ ﾁ09, ) ﾂ03 ﾁ , ﾁ n m n ﾙ m ﾁ n, f n , ﾁ f. ﾂ , f k , , 0803 af k ﾂ ﾁ , ﾁ , ﾁ ﾁ ｨｦ, , ﾁ020303: ﾁ ﾁ070109, ﾁ , ﾁ ﾁ (ｨｲ , ﾁ ﾁ , ﾁ ﾁ ﾁ ) ﾁ03 ﾁ , ﾁ ( ﾁ ﾁ09 ﾁ ﾁ ﾁ ﾁ09, ﾁ ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , , ﾁ ﾁ , , ﾁ n, f 1 07 f ﾁ f 1 +f 2 = f 3, f , n f 1 = (6ﾓ11)f 2 + f (6ﾓ11)f n, ﾁ ﾁ f 1 ﾁ ﾁ ﾁ ﾁ , ﾁ f f m 0102 ﾁ ﾁ " ﾁ , ﾂ ﾁ ﾁ h h k h i , ﾁ f 1 ; : : : ; f m. ｨ f ﾁ n, ﾁ h h k f n, g m 07 n ﾝ m f = f f k, f n, f 2 ; : : : ; f k , n; g = g g l, g m, g 2 ; : : : ; g l , m. ｨｮ ﾁ f = f f k 07 g = g 1 + +g l ﾁ , , ﾁ n. ｨｲ , f + g n n > m, ﾁ f 1. ｨｲ , ﾁ0209 n. ｨｪ , ﾁ , ﾁ ｨ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ090801, ﾁ ﾁ ﾁ ﾁ

23 ﾁ ﾁ0301, ﾁ ﾁ ﾁ ﾁ ﾁ , f n, g m f = f f k, f n, f 2 ; : : : ; f k , n; g = g g l, g m; g 2 ; : : : ; g l , m. ｨｦ : fg ﾁ f i g j ﾁ ﾁ , f 1 g ﾁ0209 m+n, ﾁ f i g j m+n , ﾁ ﾁ , ﾁ , fg, 0609 ﾁ f 1 g 1, m + n ﾁ , ﾁ , f g, ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ h 07 r, f = gh +r 07 r ﾔ r g h 07 r , 0803 h ﾁ , 09 r , , 0609 ﾁ , ﾁ , g ﾁ ﾁ gh 1 + r 1 ﾔ gh 2 + r 2 ; r 1 07 r ﾁ0905, , g g(h 1 h 2 ) ﾔ r 2 6ﾓ1 r ﾁ ﾁ ( h 1 h 2 ﾔ 0), , g, ﾁ ﾁ , , g. ｨｲ g(h 1 h 2 ) ﾔ r 2 6ﾓ1 r ﾁ , h 1 h 2 ﾔ 0 07 r 2 6ﾓ1 r 1 ﾔ , , ﾁ f 07 g ﾁ ﾁ , ﾁ h, gh 0703 ﾁ f, ﾁ090308, (ｨｲ , h ﾁ090308, ﾁ , ﾁ gh 1 07 gh ﾁ , 0803 gh 1 6ﾓ1gh ﾁ ﾁ0203 0, gh 1 07 gh gh 1 6ﾓ1 gh 2 = g(h 1 6ﾓ1 h 2 ), h 1 6ﾓ1 h ﾁ h 1 07 h h 1 6ﾓ1 h ﾁ ﾁ ｨｲ , h 1 6ﾓ1 h 2 ﾔ 0, g(h 1 6ﾓ1 h 2 ) 0703 ﾁ gh gh ﾁ ) ﾁ (070007, , ) ﾁ090801, ｨｮ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ , 09 3: ) ﾁ ﾁ , , ; 07 2) , , ; ﾁ ) , , ｨｨ ﾁ ﾁ02 ﾁ ( ﾁ ), ﾁ ﾁ ﾁ : , ﾁ , ﾁ ﾁ070103, ﾁ , ﾁ , ﾁ ﾁ ｨｦ080909, ﾁ f 07 g, , f g ﾁ h, f = gh+s, s ﾁ ﾁ , f h, gh 0703 ﾁ f h ﾁ090308, , 09 s = f 6ﾓ1gh. ｨ ﾁ090308, ﾁ ｨｦ ﾁ f 0209 g

24 ｨｲ. ｨｮ09 ﾁ ﾁ f 0209 g , ﾁ ﾁ ﾁ g ﾁ f ﾁ0203 n ﾁ ﾁ n f n h n 07 s n (n + 1) s n ﾁ , ﾁ ( ﾁ02 ﾁ0302, f g ), , m , s m ﾁ ﾁ , ﾁ02 ﾁ0302, f 0209 g ﾁ0302 s m, h 1 + h h m ｨｮ ｨｪ, ｨｪ. ｨｲ , ﾁ09 05ｨｪ ﾁ ﾁ ﾁ f 1 (cos x) 07 f 2 (cos x), f 1 07 f , 0803 f 1 (cos x) f 2 (cos x) ﾁ ﾁ03 ﾁ ｨｪ, f 1 (cos x) = f 1 (cos x)g(cos x; sin x), ﾁ ﾁ , f 1 (cos x) f 2 (cos x) ﾁ ﾁ03 ﾁ ｨｪ 07 ﾁ ﾁ , f f ｨｪ ﾁ ｨｮ03, ﾁ ﾁ a n cos nx + a n6ﾓ11 cos(n 6ﾓ1 1)x + + a 1 cos x + a, n ｨｪ ﾁ cos nx, ﾁ ｨｲ , ｨｪ f(x) 07 g(x) a n cos nx 07 b m cos mx ﾁ ﾁ , ﾁ f(x)g(x) a n b m cos nx cos mx = (a n b m =2)(cos(n + m)x + cos(n 6ﾓ1 m)x): n 07 m , (a n b m =2) cos(n + m)x ﾁ a n cos nx 07 b m cos mx, n ﾝ m, ﾁ c k cos kx, (b m cos mx)(c k cos kx) a n cos nx. ｨ : m , 0803 k = n 6ﾓ1 m; c k = 2(a n =b m ); m = 0, 0803 k = n; c k = a n =b m. ｨｲ , ﾁ , a cos n x, ﾁ ( , ｨｪ). ﾂ , : 6ﾓ1 2 cos 3x 6ﾓ1 3 cos 2x + 5 cos x cos 2x + 2 cos x cos 3x + 2 cos x + 4(cos 2x + 1) + 4 cos x 4 cos x 6ﾓ1 7 6ﾓ1 7 cos 2x 6ﾓ1 cos x 6ﾓ1 4 6ﾓ1 6ﾓ17 cos 2x 6ﾓ1 14 cos x 6ﾓ cos x + 3 ｨｦ080909, ﾁ cos x 6ﾓ1 7, cos x ﾁ ｨｪ. ﾂ ﾁ ﾁ ﾁ ﾁ09 ﾁ ｨｮ ﾁ ﾁ ﾁ ( ﾁ ). ｨｮ ﾁ ﾁ n cos n6ﾓ11 x(a cos x + b sin x), a 2 + b 2 ﾙ n > , ｨｮ ﾁ03 ﾁ ﾁ ﾁ ｨｪ ﾁ ｨｮ ﾁ03 ﾁ ﾁ ﾁ ｨｪ ﾂ ﾁ090808, ﾁ cos n6ﾓ11 x(a cos x + b sin x) 07 cos m6ﾓ11 x(c cos x + d sin x), n ﾝ m, h , ﾁ cos m6ﾓ11 x(c cos x+d sin x)h(x) 0703 ﾁ cos n6ﾓ11 x(a cos x+b sin x), n > m 07 ﾁ ﾁ e 07 f, ﾁ ec6ﾓ1fd = a 07 ed + fc = b. ｨｹ ﾁ0302 c 2 + d 2. ｨ c 2 + d 2 ﾙ 0, ﾁ ﾁ n = m h(x) h(x) , , cos n6ﾓ11 x(a cos x + b sin x) ﾔ 03 cos n6ﾓ11 x(c cos x + d sin x); c = ea; d = eb ﾁ03 ﾁ090801, ﾁ090308, a 2 + b 2 ﾙ ﾁ ﾁ ﾁ ,

25 ﾁ ﾁ f(x) = 2 cos 3 x 6ﾓ1 3 cos 2 x sin x + cos 2 x 6ﾓ1 2 sin x cos x = = cos 2 x(2 cos x 6ﾓ1 sin x) + cos x(cos x 6ﾓ1 2 sin x); g(x) = 3 cos 2 x + sin x cos x 6ﾓ1 cos x + sin x 6ﾓ1 1 = cos x(3 cos x + sin x) + (6ﾓ1 cos x + sin x) 6ﾓ1 1: ﾂ ﾁ e 07 f , (3 cosx + + sin x)(e cos x + f sin x) 0703 ﾁ cos x(2 cos x 6ﾓ1 sin x) , e 07 f , ﾁ e6ﾓ1f = 2; e+3f = 6ﾓ11. ｨｲ , e = 1=2; f = 6ﾓ1(1=2) (1=2)(cos x 6ﾓ1 sin x) cos 2 x(2 cos x 6ﾓ1 sin x) + (1=2)(1 6ﾓ1 4 sin x cos x) 6ﾓ1 (1=2)(cos x 6ﾓ1 sin x): ﾂ cos 2 x + (1=2)(cos x 6ﾓ1 sin x) + (1=2) , 0609 ﾁ , , ﾁ ﾁ ｨｮ n n > 0 a cos nx + b sin nx, a 2 + b ﾁ0203 0, ﾁ ﾂ , a cosnx + b sin nx 07 c cos mx + d sin mx 07 a 2 + b 2 ﾙ 0 07 c 2 +d 2 ﾙ h, (c cos mx+d sin mx)h(06) 0703 ﾁ a m = n, ﾁ , , h , h(x) ﾔ e, ﾁ ﾁ ﾁ ﾁ0901 a = ce; b = de n > m e 07 f, ﾁ ﾁ a cos nx + b sin nx. ｨｦ (c cos mx + d sin mx)(e cos(n 6ﾓ1 m)x + f sin(n 6ﾓ1 m)x) (c cos mx + d sin mx)(e cos(n 6ﾓ1 m)x + f sin(n 6ﾓ1 m)x) ﾔ ﾔ ce cos mx cos(n 6ﾓ1 m)x + df sin mx sin(n 6ﾓ1 m)x + + de sin nx cos(n 6ﾓ1 m)x + cf cos mx sin(n 6ﾓ1 m)x ﾔ ﾔ (1=2)ce(cos nx + cos(n 6ﾓ1 2m)x) + (1=2)df(6ﾓ1 cos nx + cos(n 6ﾓ1 2m)x) + + (1=2)de(sin nx + sin(2m 6ﾓ1 n)x) + (1=2)cf(sin nx 6ﾓ1 sin(2m 6ﾓ1 n)x): ｨｮ ｨｪ (1=2)(ce 6ﾓ1 fd) cos nx + (1=2)(ed + fc) sin nx e 07 f , ﾁ070209, ﾁ ﾁ a 07 2b, a 07 b. ｨｲ , 0102 ﾁ e 07 f. ｨｪ ﾁ , ﾁ ｨ ｨｪ 0308 ﾁ ﾁ , ﾁ ﾁ ﾁ ｨｪ ) n n ? 00) n ﾁ x n f(y), f(y) ﾁ ﾁ ?

26 ｨｲ. ｨｮ09 ﾁ ﾁ ｨｰ ﾓ ﾓ ? 3.3. ｨｰ cos 5x 0209 cos 3x (n a cos nx) ｨｰ cos nx 0209 cos mx ｨｰ cos 5x 0209 sin 3x (n a cos nx + b sin nx) ｨｰ cos nx 0209 sin mx ｨｰ cos 2x 0209 cos x + sin x n 07 m cos nx 07 sin nx 0209 cos mx + sin mx? ｨｪ sin 2x + a cos x + sin x? ｨｮ ﾁ cos 2x 07 sin 2x , ? ｨｦ03 ﾁ , f f 2 2 ﾔ 1 (f 1 07 f 2 00ｨｪ) ﾁ f 1 07 f 2? ) cos nx 07 sin nx; 00) cos nx+09 sin nx 07 cos nx6ﾓ109 sin nx, ﾁ ｨｪ ﾁ , ﾁ cosx 07 sin x ﾁ ﾁ ｨｪ ﾁ , ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ f(x 1 ; : : : ; x n ) ﾁ , k ﾁ t; x 1 ; : : : ; x n ﾁ ﾁ ﾁ03 f(tx 1 ; : : : ; tx n ) = t k f(x 1 ; : : : ; x n ) ｨｪ f(x) ﾁ0701 ﾁ ﾁ g(cos x; sin x), g , 0803 g(cos x; sin x) ﾁ ﾁ ｨｪ, f(x) ﾁ ｨｪ ﾁ ﾁ , ﾁ ﾁ070209, ﾁ ｨｮ ﾁ ｨｪ ﾁ ﾁ t, t = 2 ﾁ f(cos tx; sin tx) ﾔ t k f(cos x; sin x), ﾁ , ｨｪ f(cos 2x; sin 2x) ﾁ 02 ﾁ k f(cos x; sin x) ﾁ03 ﾁ ﾁ ﾁ , sin 2 x; cos 2 x; sin 2x; cos 2x ｨｪ (1 05 ﾁ ｨｪ , ﾔ sin 2 x + cos 2 x) , ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ a cos k x sin l x, ﾁ k +l ﾁ ﾁ0705 ﾁ ﾁ ﾁ ﾁ , f(x) = g(cos x; sin x), g , ﾁ n, 0803 g(cos x; sin x) ﾁ ﾁ a cos k x sin l x,

27 ﾁ k + l 0609 ﾁ0202 n, , , g(x; y). (ｨｲ , g ｨｪ f(x), ﾁ ﾁ ) f(x) ﾁ ﾁ ﾁ ﾁ a cos k x sin l x, k + l 0609 ﾁ0209 n 07 ﾁ k + l , n a cos k x sin l x, k + l < n, 0209 (sin 2 x + cos 2 x) m, m = (1=2)(n 6ﾓ1 (k + l)) a cos k x sin l x(sin 2 x + cos 2 x) m = h(cos x; sin x), h(y; z) = ay k z l (y 2 + z 2 ) m k + l + 2m = n ﾁ ﾁ n. ｨｮ ﾁ ｨｪ f(x) 05 ﾁ ﾁ ﾁ , ﾁ ｨｮ03 a n cos nx + a n6ﾓ11 cos(n 6ﾓ1 1)x + + a 1 cos x + a ﾁ0703 k, a 2 k + b2 k ﾁ0203 0, (a ﾁ a cos 0x + b sin 0x.) ｨｲ , ｨｪ f(x) = a n cos nx + a n6ﾓ11 cos(n 6ﾓ1 1)x + + a 1 cos x + a ﾁ ﾁ 00ｨｪ g(y), ﾁ ﾁ ﾁ c k cos kx + d k sin kx k, x = 2y ﾁ ﾂ02, 0209 ﾁ , ( ｨｮ. ｨｬ01 ﾁ03 ﾁ 縺 ﾁ), ﾁ ﾁ f(g(x); h(x)) = 0, f ﾁ , 07 ﾁ ﾁ f(0; y) = 0 07 f(1; y) = , f , f(0; y) = ﾁ , = 0. ｨ , ﾁ g(x) ﾙ f(g(x); h(x)) = ﾁ ﾁ (h(x)=g(x)) = a i, a 1 ; : : : ; a n f(1; y) = 0. ﾂ g(x) = (g(x); h(x)) = g(x) = h(x) = 0, f(0; y) = = 0 07 ﾁ g(x) = 0, f(0; y) = ﾁ ﾂ , h(x) = sin x, 09 g(x) = cos x, 0803 h(x)=g(x) tg x. ｨｦ ﾁ 0703 ﾁ , ﾁ a cos 2 x+b sin x cos x+c cos 2 x+d = 0 07 ﾁ ﾁ d 0209 d(sin 2 x+cos 2 x) , ﾁ ﾁ ﾁ f(g(x); h(x)) = 0, f(y; z) , ﾁ ﾁ ﾁ ﾁ g(y; z) 0203 ﾁ f(y; z) ﾁ ｨｪ, 0803, ﾁ ﾁ x = 2y, ﾁ f(cos 2y; sin 2y) = ﾁ , , ﾁ ｨｪ ﾁ ｨｦ080909, ﾁ ｨｪ f(cos x; sin x) ﾁ g(cos(x=2); sin(x=2)), g ｨｮ ﾁ ﾁ ﾁ f(cos x; sin x) = 0 07 ﾁ ﾁ g(cos(x=2); sin(x=2)) = g(1; tg(x=2)) = 0. ｨｹ , ﾁ ﾁ0907 sinx = 2tg(x=2)=(1 + tg 2 (x=2); cos x = (1 6ﾓ1 tg 2 (x=2))=(1 + tg 2 (x=2)) x= ﾁ ｨｪ, , cos x 07 sin x 09 tg(x=2) , tg x, ﾁ ﾁ0702 f(cos x; sin x) = 0, f(y; z) x k y l k + l, ｨｪ02 ﾁ , ﾁ ﾁ sin x 0209 tg x 07 cos x 07 ﾁ ﾁ ﾁ tg(x=2) , ﾁ ﾁ ﾂ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ0907. ｨｰ ﾁ , 0308 ﾁ ﾁ ﾁ03. ｨｪ , f 07 g 02 ﾁ ﾁ f(cos x; sin x) ﾔ ｨｦ.ｨｪ., ｨｬ01 ﾁ03 ﾁ ｨｮ.ｨｬ., ｨ.ｨｬ ｨｪ.: ｨｪ0006ｨｪ07. ｨ 07 ｨｪ ﾁ ", ｨｮ. 199.

28 ｨｲ. ｨｮ09 ﾁ ﾁ09 ﾔ g(cos x; sin x), f(cos x; sin x) 07 g(cos x; sin x) ｨｪ, ﾁ f(y; z) 07 g(y; z) ｨ f 07 g ? ﾂ , f(y; z) ﾔ g(y; z) : sin 2 x + cos 2 x ﾁ ﾁ090801, f 07 g , , f(cos x; sin x) ﾔ g(cos x; sin x) ﾁ f(y; z) ﾔ g(y; z) ﾁ , f 07 g , , f(y; z) ﾔ g(y; z), , (a; b) c a 2 + b 2 ﾙ (ka; kb) c k ﾙ 0, f(ka; kb) = g(ka; kb). ( f(a; b) = (1=k n )f(ka; kb) = (1=k n )g(ka; kb) = g(a; b), n ﾁ f 07 g.) ﾂ ﾁ (ka; kb) 0102 ﾁ ﾁ (cos '; sin '), ' [0; 2[ c cos ' = a=(a 2 + b 2 ) 1=2 ; sin ' = b=(a 2 + b 2 ) 1= ﾁ ｨｰ ﾁ x 4 6ﾓ1 3x 2 (x + 1) + 2(x + 1) 2 = 0, ﾁ y 2 6ﾓ1 3yz + 2z 2 = ｨｰ ﾁ sin 4 x 6ﾓ1 3 sin 2 x cos x + 2 cos 2 x = ｨｰ ﾁ sin 4 x 6ﾓ1 3 sin 2 x(cos x + 1) + 2(cos x + 1) 2 = ﾁ ｨｪ 0703s 4 x + sin 4 x + a(cos 2 x 6ﾓ1 sin 2 x) + b ﾁ s 4 x + sin 4 x + a(cos 2 x 6ﾓ1 sin 2 x) + b = ﾁ ｨｪ 3 sin 3 x + 4 cos x sin 2 x 6ﾓ1 2 cos 2 x sin x 6ﾓ1 3 cos 3 x sin x + 2 cos x ﾁ sin 3 x + 4 cos x sin 2 x 6ﾓ1 2 cos 2 x sin x 6ﾓ1 3 cos 3 x sin x + 2 cos x = ｨｪ f(x) = 3 sin 2x + 2(sin x 6ﾓ1 cos x) 6ﾓ ﾁ070209, ﾁ ﾁ = ﾂ , f(y; z) 07 g(y; z) m 07 n ﾁ ﾁ , m < n, ﾁ09 f(cos x; sin x) ﾔ g(cos x; sin x) , n 6ﾓ1 m f(y; z) ﾔ (y 2 + z 2 ) (n6ﾓ1m)=2 g(y; z)? = ' ﾁ [0; 2[ ｨｪ ﾁ ? ﾂ , ﾁ ﾁ ﾁ ﾁ , ﾁ0906 ﾁ ﾁ [0; 2[? ｨｮ09 ﾁ ﾁ ｨｲ ﾁ0209, ﾁ ﾂ ( ﾂｨｲｨｪ04). [email protected]

29 13ｨｪ ﾁ ｨ. ｨ. ｨｹ ﾁ ｨ ﾁ ﾁ ﾁ ﾁ ﾁ. ﾂ ( ﾁ ) ﾁ [1] ﾁ [8] 07 [11]. ﾂ [2] , ﾁ ﾁ , ﾁ ﾁ ( , ﾁ ﾁ ) , ﾁ , ﾁ , ﾁ ﾁ , 0703 ﾁ ﾁ , ﾁ ｨｲ ﾁ , ﾁ ﾁ020301, [1-10], ﾁ ｨｹ ﾁ ( , ) ﾁ ﾁ , ﾁ A ﾁ ( ) m, ﾁ , ( ) ma ( m ) m 1 A 1, m 2 A 2,..., m n A n ﾁ M, ﾁ ﾁ ﾁ ﾁ03 m 1 6ﾓ16ﾓ16ﾓ1 MA 1 + m 2 6ﾓ16ﾓ16ﾓ1 MA m n 6ﾓ16ﾓ16ﾓ1 MA n = 0: (1) , ﾁ ﾁ A 1 07 A m 1 07 m A 1 A 2 ﾁ m 2 : m 1 ( ﾁ ). ｨｲ ﾁ ﾁ ﾂ ﾁ (1) ﾁ M = c(m 1 A 1 ; m 2 A 2 ; : : : ; m n A n ) 07 m 1 A 1 + m 2 A m n A n = (m 1 + m m n )M: M = c(m 1 A 1 ; m 2 A 2 ; : : : ; m n A n ) , O (m 1 + m m n ) 6ﾓ16ﾓ1 OM = m 1 6ﾓ16ﾓ1 OA1 + m 2 6ﾓ16ﾓ1 OA2 + + m n 6ﾓ16ﾓ1 OA n : (2) ﾁ , ﾁ ﾁ , 0609 ﾁ , ﾁ , , ﾁ ﾁ , ; ﾁ ,

30 1328 ｨ. ｨ. ｨｹ ﾁ , , ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ : 1, ﾁ [ ﾂ ] ﾂ ABC ﾁ M. ｨｬ AM, BM, CM BC, CA 07 AB ﾁ ﾁ ﾁ A 1, B 1, C 1. ｨｦ03 ﾁ , AC 1 : C 1 B = p, AB 1 : B 1 C = q , AM : MA 1 = p + q : A; B; C. ｨｪ ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ A ﾁ B, B ﾁ C. ｨｪ C ﾁ A? 9. ﾂ ABC ﾁ , BC, CA, AB ﾁ ﾁ ﾁ A, B, C , AA, BB 07 CC ﾁ ｨｮ ﾁ ﾁ ﾁ , , ﾁ AB 07 BC ABC M 07 N ﾁ ﾁ , AM : MB = 3 : 5, BN : NC = 1 : CM 07 AN ﾁ O AO : ON 07 CO : OM. 12. ﾂ ABC BC 07 CA ﾁ ﾁ ﾁ A 1 07 B 1, ﾁ AB 1 : B 1 C = a 1 : a 2 07 BA 1 : A 1 C = b 1 : b AA 1 07 BB ﾁ O , ﾁ O AA 1 07 BB ﾂ ABC AB ﾁ E , AE : EB = CF : FE = k; F CE AD k ABC AC D, AD : DC = 2, BC ﾁ E 07 F , BE = EF = FC BD AE 07 AF ﾁ ﾁ ﾁ K 07 N KEFN, S ABC = BC 07 CA ABC ﾁ K 07 L; O ﾁ AK 07 BL , ﾁ OLA, OAB, OBK 0609 ﾁ ﾁ ﾁ , AB, BC 07 AC ABC ﾁ ﾁ ﾁ M, N 07 K , AM : MB = 2 : 3, AK : KC = 2 : 1, BN : NC = 1 : 2. ﾂ MK AN? 17. ﾂ ABC BC, CA 07 AB ﾁ ﾁ ﾁ A 1, B 1 07 C 1, ﾁ CA 1 : A 1 B = a 1 : a 2, AB 1 : B 1 C = b 1 : b 2 07 BC 1 : C 1 A = c 1 : c AA 1 07 B 1 C ﾁ O , ﾁ O AA 1 07 B 1 C ﾂ ABC N CB, Q AC ﾁ AQ : QC = 3 : 5, M 07 P AB ﾁ NM 07 PQ ﾁ O MO : ON. 19. ﾂ ABC ﾁ M, N, P 07 Q, CN : NB = a 1 : a 2, BM : MA = d 1 : d 2, BP : PA = c 1 : c 2, AQ : QC = b 1 : b MN 07 PQ ﾁ O = ﾊ AB m 1, m 2, m 3, O = c(m 1 A; m 2 B; m 3 C):

31 13ｨｪ ﾁ ABC , , ﾁ AB 07 AC, , AB 07 BC, ACB ﾁ ﾁ ABC ﾁ A 1 ﾊ BC, B 1 ﾊ CA, C 1 ﾊ BA AA 1, BB 1 07 CC ﾁ , CA 1 A 1 B BC 1 C 1 A AB 1 B 1 C = 1: ｨｪ ABC A 1, B 1 07 C CB, AC 07 BA ﾁ , ﾁ ﾁ ( ﾁ ﾁ ﾁ09 6ﾓ16ﾓ1 CA 1 = 6ﾓ16ﾓ1 A 1 B, 6ﾓ16ﾓ1 AB 1 = 6ﾓ16ﾓ1 B 1 C, 6ﾓ16ﾓ1 BC 1 = 6ﾓ16ﾓ1 C 1 A) A 1, B 1 07 C , = 6ﾓ , ﾁ ﾁ ﾁ , ( ﾁ ﾁ ﾁ ﾁ ﾂ , ﾁ ﾁ ﾁ ﾁ ｨｬ ) ABC; BM M ﾁ N, ﾁ , AM 07 AB ﾁ P 07 Q ﾁ ﾁ QM 07 NC ﾁ R, RB 07 AC ﾁ S ﾁ ﾁ03 PM = MS. 25. [ ] ABCD ﾁ , 09 O ﾁ ｨｬ AO, BO, CO, DO ﾁ ﾁ A 1, B 1, C 1, D ﾁ ﾁ , AO + BO + CO + DO = 3: AA 1 BB 1 CC 1 DD ﾁ ABCD ﾁ K ﾊ AB, L ﾊ BC, M ﾊ CD, N ﾊ DA, AK : KB = DM : MC = ; BL : LC = AN : ND = O ﾁ KM 07 LN , NO : OL =, KO : OM = ABCD ﾁ K ﾊ AB, L ﾊ BC, M ﾊ CD, N ﾊ DA, AK : KB = DM : MC = 07 BL : LC = AN : ND = , KM 07 LN ﾁ O, NO : OL =, KO : OM = , , , ﾁ ABCD, ﾁ K ﾊ AB; L ﾊ BC; M ﾊ CD; N ﾊ DA KM 07 LN ﾁ Z , ﾁ Z KM, AK = a; BL = b, CM = c; DN = d ﾁ D ABCD ﾁ , =n AB, ﾁ A AC ? 31. ABCD ﾁ ; Q CD; M ABCD; QM AB ﾁ K AK : KB, AD = a, BC = b ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ , ﾁ ｨｮ ﾁ , , ﾁ ﾁ

32 1330 ｨ. ｨ. ｨｹ ﾁ ﾁ , ﾁ ﾁ ABCD , M 07 N ﾁ BC 07 AD, O , E ﾁ , M, N, O 07 E ABC. ﾂ0203 ﾁ BC ﾁ T, AB 07 AC ﾁ P 07 Q ﾁ ﾁ QB 07 P C ﾁ M , A, T 07 M ﾁ SABCD ABCD ﾁ SA, SB, SC 07 SD ﾁ ﾁ ﾁ A 1, B 1, C 1 07 D 1. ｨｦ03 ﾁ , SA 1 : SA = 1 : a, SB 1 : SB = 1 : b, SC 1 : SC = 1 : c SD 1 : SD. 36. ﾂ ﾁ , ﾁ ｨｦ ﾁ ﾁ , ﾁ ﾁ 02 ﾁ , ﾁ , ﾁ , ﾁ ﾁ , ﾁ ﾁ ABC M = c(x 1 A; x 2 B; x 3 C), x 1 + x 2 + x 3 = ﾁ , M (09ｨｮ05) x 1, x 2, x 3 ( ABC). ｨ ﾁ ﾁ ﾁ , ﾁ ( ) , ﾁ , ﾁ ﾁ ﾁ0203 ﾁ O ﾁ ABC , ﾁ ﾁ03: x 1 : x 2 : x 3 = S OBC : S OCA : S OAB : , ﾁ ABC ﾁ : ax 1 + bx 2 + cx 3 = 0; ﾁ , a = b = c I ﾁ ABC, ﾁ BC ﾙ AB; D AC; B AC , B 1 I BD , ﾁ , ﾁ ABC, ﾁ 09ｨｮ ﾁ0702 a 2 x 2 x 3 + b 2 x 3 x 1 + c 2 x 1 x 2 = 0; a = BC, b = CA, c = AB ABC A 1 ﾊ BC, A 2 ﾊ A 1 C, B 1 ﾊ CA, B 2 ﾊ B 1 A, C 1 ﾊ AB, C 2 ﾊ C 1 B, CA 1 = CB 2 = a + b a b a + b + c ; AB 1 = AC 2 = b + c b c a + b + c ; BC 1 = BA 2 = c + a c a a + b + c : , A 1 C 2, C 1 B 2 07 B 1 A ABC ﾁ , ﾁ A 1 A 2 A 3 A 4 ﾁ 09ｨｮ ﾁ V ABCD ABCD ﾁ ﾁ ﾁ ﾁ ( AB; 6ﾓ1 6ﾓ1 6ﾓ1 AC; AD) ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ a, b, c a (b ﾁc), ﾁ , ﾁ ﾁ

33 13ｨｪ ﾁ , ﾁ ﾁ AB; 6ﾓ1 6ﾓ1 6ﾓ1 AC; AD ﾁ ﾁ 6ﾓ1 AB; 6ﾓ1 AC; AD 6ﾓ1 6ﾓ1 6ﾓ1 6ﾓ16ﾓ1 07 AB; AC; AM ﾁ ﾁ , M 07 D ABC. ｨｲ , ﾁ0903 ﾁ ﾁ , ﾁ ﾁ , , M(x 1 ; x 2 ; x 3 ; x 4 ) A 1 A 2 A 3 A 4 ﾁ x 1 = V MA 2 A 3 A 4 V A1 A 2 A 3 A 4 ; x 2 = V A 1 MA 3 A 4 V A1 A 2 A 3 A 4 ; x 3 = V A 1 A 2 MA 4 V A1 A 2 A 3 A 4 ; x 4 = V A 1 A 2 A 3 M V A1 A 2 A 3 A 4 : ABCD. ｨｦ03 ﾁ : S BCD = S 1, S ACD = S 2,S ABD = S 3, S ABC = S ﾁ ﾁ ﾁ0203 ﾁ [ ｨｮ0800, 2011.] , ﾁ ﾁ0203 ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ0203 ﾁ [ ] ﾁ ﾁ0701 ﾁ , ﾁ ﾁ ﾁ03 ﾁ , ﾁ ﾁ 0609 ﾁ ﾂ ﾁ ﾁ [ ﾁ ] ﾁ ﾁ0701 ﾁ , ﾁ ﾁ ﾁ03 ﾁ , ﾁ ﾁ 0609 ﾁ , ﾁ , ﾁ , ﾁ , ﾁ [ｨｪｨｪ07, 2014, ﾁ , 11-5.] 0803 ﾁ ﾁ A 1 B 1 C 1 A 2 B 2 C ﾁ A i B j C k, i; j; k = 1; 2. ｨｮ ﾁ O ﾁ , O ﾁ A 1 A 2, B 1 B 2 07 C 1 C ﾁ ﾁ AO : ON = 3 : 4; CO : OM = 32 : AO : OA 1 = a 1 (b 1 + b 2 ) : a 2 b 1 ; BO : OB 1 = b 1 (a 1 + a 2 ) : a 1 b ﾌ : AO : OA 1 = (a 1 + a 2 )b 1 c 2 : (a 1 b 1 c 1 + a 2 b 2 c 2 ); B 1 O : OC 1 = a 1 b 1 (c 1 + c 2 ) : a 2 c 2 (b 1 + b 2 ). 18. MO : ON = 6 : m 1 = a 1 b a 2 b 2 c 2 d 1 c 2 d 1 6ﾓ1 c 1 d ; m 2 = a 1b a 2 b 2 c 2 d 2 2 c 2 d 1 6ﾓ1 c 1 d ; m 3 = a 2 b 1 : KZ : ZM = ab(c + d) : cd(a + b). 1 a n + 1 b a=p, b=p, c=p, a, b, c , P a 6ﾓ1 b + c ﾁ0203 ﾁ , a, (6ﾓ1a), b, c. 44. ﾆ A i A 2 jx i x j = ﾁ x i = S i =S, S ﾁ ﾁ0203 ﾁ ﾁ ﾁ ﾁ ﾁ

34 1332 ｨ. ｨ. ｨｹ ﾁ ｨｰ ﾁ ﾁ ﾁ O r = 6ﾓ16ﾓ1 OM 07 r i = 6ﾓ16ﾓ1 OM i i = 1; 2; : : : ; n ﾆ 6ﾓ16ﾓ1 mi MA i = ﾆ ﾆ m i (r i 6ﾓ1 r) = 0 6ﾌ86ﾍ0 r mi r = ﾆ i : mi , ﾁ ﾁ M ﾁ ﾁ ﾁ ﾁ ﾂ 0609 ﾁ ﾁ03 (2) ﾁ ﾁ , 0308 ﾁ , ( ﾁ ﾁ ﾁ09, ﾁ09) ABCD , E, F, G, H, K, N ﾁ ﾁ ﾁ AB, BC, CD, DA, AC, BD. ｨｰ ﾁ A, B, C, D ﾁ ﾁ ﾁ ﾁ M , 0803 M = c(1a; 1B; 1C; 1D) = c(2e; 2G) = c(2f; 2H) = c(2k; 2N): ﾁ0907 ﾁ E 07 G, F 07 H, K 07 N ﾁ , M ﾁ EG, FH 07 KN. 5. ﾂ ﾁ , A, B, C 07 D ﾁ , A, B, C, D ﾁ ABCD ﾁ , A BCD. ｨｰ ﾁ ﾁ B + 1C + 1D = 3A 1 07 M = c(1a; 1B; 1C; 1D) = c(1a; 3A 1 ). ｨｲ ﾁ AA 1 ﾁ : 1, ﾁ A ﾁ ﾁ ! 7. ｨｲ , C 1 = c(1a; pb), 09 B 1 = c(1a; qc) (1A; pb; qc) ﾁ AA 1 07 BB 1. ｨｹ , M = c(1a; pb; qc) pb + qc = (p + q)a 1 07 M = c(1a; (p + q)a 1 ) ﾁ ﾁ , , ﾁ ﾂ ﾁ ﾁ : x = AB = AC, y = BA = BC, z = CB = CA M = c(yza; zxb; xyc) M = c(y(x + z)b ; zxb) ﾊ BB ﾁ , M AA 07 CC. ｨｲ ｨｹ M ﾁ , ﾁ ( 0703 ﾁ A, B, C 07 D 0609 ﾁ ﾁ ﾁ a, b, c 07 d a A; 1 b B; 1 c C; 1 ) d D , ﾁ ﾁ ｨｲ , , ﾁ ﾁ B ﾁ m 2 ﾙ 0. ｨｪ m 1 07 m ﾁ0708 N = c(m 2 B; m 3 C) 07 M = c(m 1 A; m 2 B) ﾁ ﾁ ﾁ09 m 2 = 4m m 1 = 5m 2.

35 13ｨｪ ﾁ ｨｰ m 2 = m 3 = 3, m 1 = O = c(20a; 15N), AO : ON = 15 : 20 = 3 : ﾁ O = c(3c; 32M). ｨｲ , CO : OM = 32 : ﾁ ｨｰ ﾁ ﾁ , ﾁ O. ﾂ , B 1 = c(m 1 A; m 3 C); (3) 0803, ﾁ m 1 A 07 m 3 C , ﾁ ﾁ m 2 B 07 (m 1 + m 3 )B , BB , c(m 2 B; m 3 C) = A 1 ; (4) 0803 c(m 1 A; m 2 B; m 3 C) ﾊ AA O = AA 1 ﾉ BB 1, , O = c(m 1 A; m 2 B; m 3 C) ﾁ0705 (3) 07 (4), ﾁ , ﾁ ﾁ m 1 a 1 = m 3 a 2 ; m 2 b 1 = m 3 b 2 : (5) , ﾁ ﾁ ﾁ ﾁ03 m 3 ﾁ ﾁ , ﾁ ﾁ (5) 02 ﾁ m 3 = a 1 b m 1 = a 2 b 1 07 m 2 = a 1 b , , ﾁ ﾁ O = c(a 1 b 2 B; (a 2 b 1 +a 1 b 1 )B 1 ), ﾁ BO : OB 1 = b 1 (a 1 +a 2 ) : a 1 b 2. ｨ , AO : OA 1 = a 1 (b 1 + b 2 ) : a 2 b ｨｰ ﾁ ﾁ , ﾁ F , , ﾁ A 07 B, ﾁ E, 09 ﾁ C 07 B ﾁ D AE : EB = k, ﾁ ﾁ B ﾁ k , ﾁ A ﾁ ﾁ ﾁ C 07 K. ｨｦ080909, F = c(1a; kb; kc) = c((1 + k)e; kc) ﾁ0705 CF : FE = k, ﾁ , k 2 = k +1, k , 0308 ﾁ , ﾁ , , ﾁ K 07 N AE 07 AF, S KEFN = ( 1 6ﾓ1 AK AE AN AF ) S AEF ; S AEF = 1 3 S ABC: ｨｲ , N = c(1a; 2C; 1B) = c(1a; 3F) AN = 3 4 AF. ｨｮ , K = c(1a; 2C; 4B) = c(1a; 6E) 07 AK = 6 7 AE.

36 1334 ｨ. ｨ. ｨｹ ﾁ ｨｲ , BO : OL = S BOA : S OLA = 6 : 5 07 AO : OK = S AOB : S KOB = 6 : x, y 07 z, O = c(xa; yb; zc) , L = c(xa; zc) 07 K = c(yb; zc) O = c(yb; (x + z)l) 07 BO : OL = 6 : 5, y = 5(x + z). ｨ O = c(xa; (y + z)k) 07 AO : OK = 6 : x = 7(y + z). ｨｦ ﾁ , x = 77z 07 y = 65z CL = 77LA, S BCL = 77S ALB, S ABC = 78S ALB = = 858: , O = c(m 1 A; m 2 B; m 3 C). ｨｲ , O ﾊ AN 6ﾌ86ﾍ0 N = c(m 2 B; m 3 C). ｨｲ , m 2 = 2m KM, m ﾁ m 1 07 m 茖 , M = c(m 1A; m 2 B) 07 K = c(m 茖 1; m 3 C) m 1 = 3m m 茖 1 = m m 3 = m 2 = 4, m 1 = 6, m 茖 1 = ﾁ O AN 07 KM. ｨｰ ｨｦ O = c(7a; 6N). ｨｲ , AO : ON = 6 : ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ , ﾁ O. ｨｪ ﾁ ﾁ B 07 C ﾁ ﾁ0705 A 1 = c(m 2 B; m 3 C): (6) ｨｹ ﾁ ﾁ AA 1. ｨ ﾁ , ﾁ B 1 C 1, ﾁ ﾁ A, , ﾁ ﾁ ﾁ m 1A 07 m 茖 1A, ﾁ ﾁ0907 B 1 = c(m 茖 1A; m 3 C); C 1 = c(m 1A; m 2 B): (7) ｨｦ03 (6) 07 (7) ﾁ ﾁ02 ﾁ ﾁ ﾁ09 m 2 a 2 = m 3 a 1 ; m 1c 2 = m 2 c 1 ; m 茖 1b 1 = m 3 b 2 : ﾁ m 2 = a 1 b 1 c m 3 = a 2 b 1 c 2, m 1 = a 1 b 1 c 1, m 茖 1 = a 2 b 2 c O = c((m 1 + m 茖 1)A 1 ; m 2 B + m 3 C) = c((a 1 b 1 c 1 + a 2 b 2 c 2 )A; (a 1 b 1 c 2 + a 2 b 1 c 2 )A 1 ); AO OA 1 = (a 1 + a 2 )b 1 c 2 a 1 b 1 c 1 + a 2 b 2 c 2 :

37 13ｨｪ ﾁ ﾁ O = c(m 1A + m 2 B; m 茖 1A + m 3 C) = c((a 1 b 1 c 1 + a 1 b 1 c 2 )C 1 ; (a 2 b 2 c 2 A + a 2 b 1 c 2 )B 1 ) B 1 O OC 1 = a 1b 1 (c 1 + c 2 ) a 2 c 2 (b 1 + b 2 ) : (xa; yb; zc), ﾁ O ( ). ｨｰ ｨｰ ﾁ ﾁ A (x = x 1 + x 2 ) , Q = c(x 1 A; zc) 07 P = c(x 2 A; yb) ﾁ ﾁ ﾁ , x 1 = 5 3 z 07 x 2 = 1 2 y ﾁ ﾁ B (y = y 1 + y 2 ) , M = c(y 1 B; xa) 07 N = c(y 2 B; zc) y 1 = 1 2 x 07 y 2 = z. ｨｦ ﾁ x = 5 3 z y; y = 1 2 x + z: z = x = 26 3, y = , ﾁ O MN, , ﾁ ﾁ ﾁ0309 y 1 B +xa 07 y 2 B +zc y 1 = x 2 = 13 3, 09 y 2 = z = 3, B + 26 A = 13M; 3B + 3C = 6N: , O = c(13m; 6N). ｨｲ , MO : ON = 6 : (xa; yb; zc). ｨｰ

38 1336 ｨ. ｨ. ｨｹ ﾁ ｨｰ x = x 1 + x , P = c(x 1 A; yb) 07 Q = c(x 2 A; zc) ﾁ ﾁ ﾁ09 x 1 c 2 = yc 1 07 x 2 b 1 = zb PQ. ｨ MN ﾁ ﾁ B: y = y 1 + y 2 ; c(y 1 B; xa) = M; c(y 2 B; zc) = N; y 1 d 1 = xd 2 ; y 2 a 2 = za 1 : ｨｮ : x = x 1 + x 2 = c 1 c 2 y + b 2 b 1 z; y = y 1 + y 2 = d 2 d 1 x + a 1 a 2 z: ﾁ z = a 2 b 1 ( ﾁ ﾁ , , ) ﾁ ﾁ ﾁ ﾁ 02 ﾁ ﾁ : x = c 1 y + a c 2 b 2 ; y = d 2 x + a 2 d 1 b 1 : (8) 1 ｨｦ ﾁ , M 07 P ﾁ BA ﾁ : 1 c 2 ﾙ 1 d 2. ｨｬ ﾁ , (8) ﾁ x = a 1b 1 c 1 d 1 + a 2 b 2 c 2 d 1 c 2 d 1 6ﾓ1 c 1 d 2 ; y = a 1b 1 c 2 d 1 + a 2 b 2 c 2 d 2 c 2 d 1 6ﾓ1 c 1 d 2 : ｨｲ ﾁ [2], ﾁ ﾁ : x + y + x = 0 6ﾌ86ﾍ0 MN ｬ P Q. ｨｲ , ﾁ , ﾁ ﾁ : ﾁ ﾁ , ﾁ , , ｨｰ , , a = BC, b = CA, c = AB. ｨｪ , l C AB ﾁ b : a, ﾁ A ﾁ ﾁ A a, 09 ﾁ ﾁ B b, z c(aa; bb; zc) l MN, M 07 N AB 07 AC, b b = a + z, z = b 6ﾓ1 a. ｨｲ , E = l ﾉ MN, 0803 E = c(aa; bb; (b 6ﾓ1 a)c) P 07 Q ﾁ AB 07 AC ( ) ﾁ , AP = AQ = p 6ﾓ1a, BP = p 6ﾓ1b, CP = p 6ﾓ1c, p ABC E PQ, a ﾁ ﾁ ﾁ a = a 1 + a 2, c(a 1 A; bb) = P 07 c(a 2 A; (b 6ﾓ1 a)c) = Q ﾁ a 2 (p 6ﾓ1 a) = b(p 6ﾓ1 b); a 1 (p 6ﾓ1 a) = (b 6ﾓ1 a)(p 6ﾓ1 c):

39 13ｨｪ ﾁ ｨｲ , ﾁ ﾁ ﾁ03 a = p 6ﾓ1 b p 6ﾓ1 a b + p 6ﾓ1 c (b 6ﾓ1 a): p 6ﾓ1 a ｨｦ , ﾁ ﾁ , ﾁ ﾁ ｨｦ080909, ﾁ ﾁ O = CC 1 ﾉ AA 1. ｨｰ ﾁ ﾁ ABC , ﾁ O a 1 = AB 1, a 2 = B 1 C, b 1 = BC 1, b 2 = C 1 A, c 1 = AB 1, c 2 = B 1 C. ｨｬ ﾁ , (a 1 b 1 A; a 1 b 2 B; a 2 b 2 C) ( ). ｨｰ ﾁ ﾁ ﾁ03 CA 1 A 1 B BC 1 C 1 A AB 1 = 1; (9) B 1 C 0803 a 1 b 1 c 1 = a 2 b 2 c 2 07 c(a 1 b 1 A; a 2 b 2 C) = B O BB 1, ﾁ090107, AA 1, BB 1 07 CC ﾁ , BB O, 0803 B 1 = c(a 1 b 1 A; a 2 b 2 C), ﾁ0203 ﾁ ﾁ ﾁ03 a 1 b 1 c 1 = a 2 b 2 c 2, 0609 ﾁ (9) z z 1 07 z , c(z 1 C; xa) = B 1 07 c(z 2 C; yb) = A 1, 0803 c(xa; yb; zc) ﾊ A 1 B 1. ﾂ D = c(xa; yb; zc) ﾊ A 1 B m 07 m 茖, D = c(m A 1 ; m 茖 B 1 ) 07 m + m 茖 = x + y + z x 07 y z 1 07 z 2, c(z 1 C; xa) = B 1 07 c(z 2 C; yb) = A z 1 + x = m 茖, z 2 + y = m, ﾁ02 ﾁ , z = z 1 + z 2. ｨｦ080909, (xa; yb; zc) A 1 B , ﾁ z = z 1 + z 2, c(z 1 C; xa) = B 1 07 c(z 2 C; yb) = A 1, ﾓ16ﾓ1 z 1B1 C + x 6ﾓ16ﾓ1 6ﾓ16ﾓ1 B 1 A = 0; z 2A1 C + y 6ﾓ16ﾓ1 A 1 B = 0: (10) ﾁ ﾓ16ﾓ1 AB 1 = 6ﾓ16ﾓ1 B 1 C 07 6ﾓ16ﾓ1 CA 1 = 6ﾓ16ﾓ1 A 1 B, , ﾁ0705 (10) ﾁ ﾁ z 1 = x 07 z 2 = y. ｨｦ ﾁ ﾁ : ｨ ﾁ : c(xa; yb; zc) ﾊ A 1 B 1 6ﾌ86ﾍ0 z = x + y : (11) c(xa; yb; zc) ﾊ B 1 C 1 6ﾌ86ﾍ0 x = y + z : (12)

40 1338 ｨ. ｨ. ｨｹ ﾁ A 1, B 1 07 C , ﾁ A 1 B 1 07 B 1 C 1, , (11){(12), ﾁ ﾁ ﾁ ﾁ09 z = x + y ; x = y + z : z = (y + z ) + y = y + z + y, = 0, = 6ﾓ11, ﾁ M ﾁ AA 1, BB 1 07 CC m 1, m 2, m 3 ﾁ , M = c(m 1 A; m 2 B; m 3 C) A 2, B 2, C ﾁ ﾁ BC, CA, AB, 09 A 3, B 3, C ﾁ ﾁ ﾁ O = c(m 1 (m 2 + m 3 )A; m 2 (m 1 + m 3 )B; m 3 (m 1 + m 2 )C). ｨｲ , m 1 (m 2 + m 3 )A + m 2 (m 1 + m 3 )B + m 3 (m 1 + m 2 )C = = m 1 (m 2 + m 3 )A + m 1 (m 2 B + m 3 C) + m 2 m 3 (1B + 1C) = = m 1 (m 2 + m 3 )A + m 1 (m 2 + m 3 )A 1 + 2m 2 m 3 A 2 = = 2m 1 (m 2 + m 3 )A 3 + 2m 2 m 3 A 2 : ｨｲ , O A 2 A 3. ｨ ﾁ , O B 2 B 3 07 C 2 C ﾁ A 07 C , 09 ﾁ B 07 N x 07 y, Q = c(1a; xb) 07 M = c(xb; yn) ( ). ｨｰ M = c(1a; 1C), M = c(1a; xb; 1C; yn) = c((1 + x)q; 1C; yn): R = c(1c; yn) ﾁ A, B 07 N, c(1a; xb; yn) = c((1 + x)q; yn) ﾊ QN; c(1a; xb; yn) = c(1a; (x + y)m) ﾊ AM; c(1a; xb; yn) = QN ﾉ AM = P. ｨ ﾁ (1C; xb; yn) ﾁ c(1c; xb; yn) = c((1 + y)r; xb) ﾊ BR; c(1c; xb; yn) = c(1c; (x + y)m) ﾊ CM; c(1c; xb; yn) = BR ﾉCM = S , ﾁ (1A; 1C; 2xB; 2yN). ｨｮ , M = c(1a; 1C) 07 M = c(2xb; 2yN), M = c(1a; 1C; 2xB; 2yN). ｨｮ , 1A+xB +yn = (1+x+y)P 07 1C +xb +yn = (1+x+y)S. ｨｲ , M = c ( (1+x+y)P,

41 13ｨｪ ﾁ (1 + x + y)s ) , ﾁ ﾁ ﾁ P 07 S ﾁ M M PS O = c(m 1 A; m 2 B; m 3 C; m 4 D), m 1 + m 2 + m 3 + m 4 = ﾁ , A 1, B 1, C 1, D , ﾁ , m 2 B + m 3 C + m 4 D = (m 2 + m 3 + m 4 )A 1 = (1 6ﾓ1 m 1 )A 1 : O = c(m 1 A; (1 6ﾓ1 m 1 )A 1 ), ﾁ m 1 AO = (1 6ﾓ1 m 1 ) OA 1 ; m 1 AO = (1 6ﾓ1 m 1 ) (AA 1 6ﾓ1 AO), AO AA 1 = 1 6ﾓ1 m 1. ｨｮ ﾁ ﾁ ﾁ09, AO + BO + CO + DO = 1 6ﾓ1 m AA 1 BB 1 CC 1 DD ﾓ1 m ﾓ1 m ﾓ1 m 4 = 4 6ﾓ1 1 = 3: ｨｦ c(1a; B) = K, c(d; C) = M , ｨ , c(1a; B; C; D) ﾊ LN. ｨｲ , c(1a; B; C; D) ﾊ KM: O = c(1a; B; C; D) = c((1 + )K; (1 + )M) = c((1 + )N; (1 + )L): KO : OM =, NO : OL =. 27. ﾂ ﾁ , A, B, C 07 D ﾁ , A, B, C, D ﾁ ﾁ ﾁ O , ﾁ ﾁ r 1, r 2, r 3, r 4 ﾁ , O 09 ﾁ ﾁ ; , ﾁ ﾁ ﾁ ｨｰ ﾁ ﾁ ( ) m 1, m 2, m 3, m 4, O m 1 r 1 + m 2 r 2 + m 3 r 3 + m 4 r 4 = 0: ﾁ ﾁ ﾁ ﾁ r ﾁ ﾁ (0003 ﾁ , ﾁ ﾁ ) ﾁ ﾁ , ﾁ ﾁ ﾁ ( 1 Z = c a A; 1 b B; 1 c C; 1 ) (( d D 1 = a + 1 ) ( 1 K; b c d) + 1 ) M : KZ ZM = 1 c + d 1 1 = a + 1 b ab(c + d) cd(a + b) : ﾁ AB ﾁ K, AC ﾁ N O ｨｲ , N = c((n 6ﾓ1 1)A; 1B; 1D) = c((n 6ﾓ1 1)A; 2O) AN : AO = 2 : (n + 1) 07 AN : AC = 1 : (n + 1) MD = x, MA = z, MB = y, MC = t. ｨｦ ﾁ ADM 07 BCM x : t = z : y = a : b. ｨｰ ﾁ ﾁ

42 1340 ｨ. ｨ. ｨｹ ﾁ , ﾁ M ﾁ C 07 D ﾁ ｨｦ M = c(yzc; yta) = c(yzd; xzb) K = c(yta; xzb) 07 AK KB = xz yt = x t z y = a b a b = a2 b 2 : A 1 ; A 2 ; : : : ; D 3 ; D 4 ( ). ｨｰ ｨｮ , S A1 B 1 B 2 A 2 = 1 3 S ABCD H, H 1, H ﾁ ﾁ, B, B 1 07 B AD H 2 HBB , 09 H 1 B B 1 H 1 = 1 2 (BH + B 2 H 2 ) 07 S A1 B 1 A 2 = 1 2 (S ABA 1 + S A2 B 2 D) S A2 B 1 B 2 = 1 2 (S A 1 BB 1 + S DB2 C). ｨｮ ﾁ ﾁ ﾁ09, S A1 B 1 B 2 A 2 = 1 2 (S ABB 1 A 1 + S B2 CDA 2 ), ﾁ ｨｦ , ﾁ ﾁ ﾁ K 07 L, N 07 M A 1 B 1 07 A 2 B ﾁ S KLMN = 1 3 S A 1 B 1 B 2 A 2 = 1 9 S ABCD: ﾁ a = AD, b = BC. ｨｦ ﾁ BOC 07 DOA O = c(ac; ba) = c(ab; bd) O = c(ba; ab; ac; bd) = c(2am; 2bN) ﾊ MN: ｨｮ , E = c(6ﾓ1ab; ba) = c(6ﾓ1ac; bd) 07 E = c(ba; 6ﾓ1aB; 6ﾓ1aC; bd) = c(6ﾓ12am; 2bN) ﾊ MN: a = AP = AQ, b = BP = BT, c = CQ = CT AB = a 6ﾓ1 b, AC = a 6ﾓ1 c ﾁ A, P, Q, ﾁ M. ｨｦ c(bca; c(a 6ﾓ1b)P) = B, c(bca; b(a 6ﾓ1c)Q) = C c(bca; c(a 6ﾓ1b)P; b(a 6ﾓ1c)Q) = QB ﾉPC = M bca+c(a 6ﾓ1b)P +b(a 6ﾓ1c)Q = (ab+ac 6ﾓ1bc)M. ｨｮ , bca+c(a 6ﾓ1b)P = acb, bca + b(a 6ﾓ1 c)q = abc, acb + abc = a(b + c)t , , bcA + c(a 6ﾓ1 b)p + b(a 6ﾓ1 c)q = bca + (ab + ac 6ﾓ1 bc)m = a(b + c)t: , ( ) ﾁ A 07 M ﾁ T. ｨｹ , T AM ﾁ0708, 1A + (a 6ﾓ1 1)S = aa 1 ; 1B + (b 6ﾓ1 1)S = bb 1 ; 1C + (c 6ﾓ1 1)S = cc 1 :

43 13ｨｪ ﾁ , ABCD , 1A + (6ﾓ11)B + 1C = 1D M = c(aa 1 ; 6ﾓ1bB 1 ; cc 1 ) M ﾊ. ｨｮ , (a 6ﾓ1 b + c)m = 1A + (a 6ﾓ1 1)S 6ﾓ1 (1B + (b 6ﾓ1 1)S) + 1C + (c 6ﾓ1 1)S = = (1A 6ﾓ1 1B + 1C) + (a 6ﾓ1 b + c 6ﾓ1 1)S = 1D + (a 6ﾓ1 b + c 6ﾓ1 1)S: ｨｲ , M DS , M = ﾉ DS = D D 1 = c(1d; (a 6ﾓ1 b + c 6ﾓ1 1)S), ﾁ ﾁ ﾁ ﾁ03 SD 1 D 1 D = 1 a 6ﾓ1 b + c 6ﾓ1 1, SD 1 SD = 1 a 6ﾓ1 b + c A 1 A 2 A ｨｰ M = c (6ﾓ1 1 r I; 1 I r 1 ; 1 I 1 r 2 ; 1 ) I 2 r 3 ; I 07 r ﾁ , I i 07 r i , ﾁ ﾁ i (i = 1; 2; 3) B i , ﾁ ﾁ , i , , B 1 = c ( 1 r2 I 2 ; 1 r 3 I 3 ) ﾁ02- ﾁ , A 1 = c ( 6ﾓ1r 1 1 I 1 ; 1 r I). ｨｲ , M ﾁ , ﾁ A 1 07 B 1. ｨｮ ﾁ , M ﾊ A 1 B ﾁ , M A 2 B 2 07 A 3 B ABC a = BC, b = CA, c = AB ﾁ I AA I = c(xa; yb; zc), 0803 A 1 = c(yb; zc) y : z = b : c. ｨ ﾁ x : y = a : b , ﾁ I 07 I ﾁ ﾁ ﾁ ﾁ0203 ﾁ , BC; 09 r 07 r ｨｨ ﾁ , r 1 = b a + c b 6ﾓ1 + a c, a = BC, b = CA, c = AB , I = c(aa; bb; cc), (a + b + c) 6ﾓ1 AI = b 6ﾓ1 AB + c 6ﾓ1 AC: I A, ﾁ x, I 1 = c(xa; bb; cc) (x + b + c) 6ﾓ16ﾓ1 AI 1 = b AB 6ﾓ1 6ﾓ1 + cac: ｨｲ , (a + b + c) 6ﾓ1 AI = (x + b + c) 6ﾓ16ﾓ1 AI 1 : (13) I 07 I AB, ﾓ1 6ﾓ16ﾓ , AI : AI 1 = r : r 1, r 1AI = rai1, ( ﾁ0905 ﾁ ﾁ r1 r ) (a + b + c) 6ﾓ1 AI = (b + c 6ﾓ1 a) 6ﾓ16ﾓ1 AI 1 : (14) ｨｦ03 (13) 07 (14) , x = 6ﾓ1a O ﾁ AA, BB, CC. ｨｲ , S OAB S OBC = S B AB S B 脾C = d A dc = AB B C ; d A 07 d C ﾁ, A 07 C BB O = c(xa; yb; zc), 0803 B = c(xa; zc) x : z = B C : AB = S OBC : S OAB. ｨ ﾁ ﾁ x : y = S OBC : S OCA :

44 1342 ｨ. ｨ. ｨｹ ﾁ ｨｰ ﾁ M(x 1 ; x 2 ; x 3 ), ﾁ ﾁ ﾁ ax 1 + bx 2 + cx 3 = 0. ｨｨ ﾁ ﾁ ﾁ b ﾙ c , ﾁ0905, x 3 = 1 6ﾓ1 x 1 6ﾓ1 x 2, ﾁ (a 6ﾓ1 c)x 1 + (b 6ﾓ1 c)x 2 + c = 0; ﾁ02 ﾁ ﾁ x x 1 : x 2 = l 1 x 1 + m 1 : ｨｦ x 3 = 16ﾓ1x 1 6ﾓ1x 2 = l 2 x 1 +m ﾁ0703 M = c(x 1 A; x 2 B; x 3 C) 0609 ﾁ ﾁ ﾁ ﾁ09 6ﾓ16ﾓ1 6ﾓ1 6ﾓ1 6ﾓ1 OM = x 1OA + x2ob + x3oc: ｨｮ , 6ﾓ16ﾓ1 6ﾓ1 OM = x 1OA + (l1 x 1 + m 1 ) OB 6ﾓ1 + (l2 x 1 + m 2 ) OC 6ﾓ1 = x1 r 0 + s; r 0 07 s ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , l A, 09 l AB 07 AC ﾁ 02 ﾁ K 07 N M ﾊ l, z 1 07 z ﾁ ﾁ03 M = c(z 1 K; z 2 N) z 1 07 z x 1, x 2, x 茖 1, x 3, z 1 K = x 2 B + x 1A 07 z 2 N = x 3 C + x 茖 1A, x 1 = k 1 x 2, x 茖 1 = k 2 x 3, k 1 07 k , ﾁ K 07 N ﾁ ﾁ AB 07 AC , k 1 ﾙ 6ﾓ11 07 k 2 ﾙ 6ﾓ11, ｨｲ , M = c((x 1+x 茖 2)A; x 2 B; x 3 C) 07 x 1 = x 1 + x 茖 1 = k 1 x 2 + k 2 x 3, ﾁ ﾁ x 1 6ﾓ1 k 1 x 2 6ﾓ1 k 2 x 3 = 0; ﾁ x , , x ﾁ ABC. ｨｲ M(x 1 ; x 2 ; x 3 ) , M = c(x 1 A; x 2 B; x 3 C) 07 x 1 +x 2 +x 3 = 1. ｨｦ ﾁ : ) ; I A(1; 0; 0); B(0; 1; 0); C(0; 0; 1); B 1 ( 1 2 ; 0; 1 2 ( a 2p ; b 2p ; c 2p ) ; D ( p 6ﾓ1 c b , , a = BC, b = CA, c = AB, p = a+b+c ﾁ BD: ; 0; p 6ﾓ1 a ) ; b (p 6ﾓ1 a)x 1 = (p 6ﾓ1 c)x 3 : (15) x 1 +x 2 +x 3 = ﾁ B 1 I ﾁ ﾁ07 ﾁ ﾁ ﾁ B 1 07 I, = 0; a + b + c = 0: (a 6ﾓ1 c) + b = = b, = c 6ﾓ1 a 07 = 6ﾓ1b. ｨｦ080909, ﾁ B 1 I: bx 1 + (c 6ﾓ1 a)x 2 6ﾓ1 bx 3 = 0: (16) , x 1 + x 2 + x 3 = 1: (17) ｨｰ ﾁ ﾁ (15){(17) ﾁ , BD 07 B 1 I: x 1 = p 2b 6ﾓ1 c, x 2 = 1 2, x 3 = p 2b 6ﾓ1 a. ｨｰ09 ﾁ ﾁ03 x 2 = ﾁ , BD.

45 13ｨｪ ﾁ ﾂ ﾁ ﾁ r 1 = 6ﾓ1 OA, r 2 = OB, 6ﾓ1 r3 = OC, 6ﾓ O ｨｲ , ﾁ ﾁ 0609 ﾁ0202 R ﾂ x 1, x 2, x M M ﾁ ﾁ ﾁ09 6ﾓ16ﾓ1 OM = x 1 r 1 +x 2 r 2 +x 3 r 3 ﾁ0303 ﾁ ﾁ 09 ﾁ : ｨｲ , ﾓ16ﾓ1 OM 2 = (x x x 2 3)R 2 + 2x 1 x 2 r 1 r 2 + 2x 2 x 3 r 2 r 3 + 2x 3 x 1 r 3 r 1 : x x x 2 3 = (x 1 + x 2 + x 3 ) 2 6ﾓ1 2(x 1 x 2 + x 2 x 3 + x 3 x 1 ) = 1 6ﾓ1 2(x 1 x 2 + x 2 x 3 + x 3 x 1 ): 6ﾓ16ﾓ1 OM 2 = R 2 6ﾓ1 (2R 2 6ﾓ1 2r 1 r 2 )x 1 x 2 6ﾓ1 (2R 2 6ﾓ1 2r 2 r 3 )x 2 x 3 6ﾓ1 (2R 2 6ﾓ1 2r 3 r 1 )x 2 x 3 : (18) ｨｪ , c = 6ﾓ1 AB = r 2 6ﾓ1r 1, c 2 = (r 2 6ﾓ1r 1 ) 2 = 2R 2 6ﾓ1 2r 2 r 1. ｨ ﾁ ﾁ09 a 2 = 2R 2 6ﾓ1 2r 3 r 2 07 b 2 = 2R 2 6ﾓ1 2r 1 r (18) ﾁ ﾁ OM 2 = R 2 6ﾓ1 (c 2 x 1 x 2 + a 2 x 2 x 3 + b 2 x 3 x 1 ): ｨｲ , M(x 1 ; x 2 ; x 3 ) ABC , c 2 x 1 x 2 + a 2 x 2 x 3 + b 2 x 3 x 1 = 0: ﾁ , ﾁ , ﾁ ﾁ0702 a 2 yz + b 2 xz + c 2 xy = 0: (19) O = B 2 C 1 ﾉ A 1 C 2 ( ). ｨｰ ｨｰ x = x 1 + x , c(x 1 A; yb) = C 1 07 c(x 2 A; zc) = B 2, y = y 1 + y , c(y 1 B; xa) = C 2 07 c(y 2 B; zc) = A 1 : ｨｹ ﾁ090203, O ﾁ B 2 C 1 07 A 1 C 2. ｨｦ ﾁ , : x = a + b c z + a + c b y; y = a + b c z + b + c x: (20) a ﾁ z = c 2, 0803, ﾁ (20), x = 6ﾓ1a(a+b), y = 6ﾓ1b(a+b) ﾁ ﾁ x, y 07 z ﾁ (19) ﾁ090308, (19) ﾁ ﾁ ｨｮ ﾁ , ﾁ ﾁ ﾁ0907 ﾁ

46 1344 ｨ. ｨ. ｨｹ ﾁ ﾁ ﾁ , ﾁ ﾁ M(x 1 ; x 2 ; x 3 ; x 4 ) : OM 2 = R 2 6ﾓ1 ﾆ i<j A i A 2 jx i x j ; O , R ｨｲ , ﾁ ﾁ03: ﾆ i<j A i A 2 jx i x j = 0: O ﾁ ﾁ ﾁ ﾁ03 6ﾓ16ﾓ1 OM = ﾁ03 O A 4, , ﾓ16ﾓ16ﾓ1 A 4 M = 3 ﾆ i=1 x i 6ﾓ16ﾓ16ﾓ1 A 4 A i : ﾁ ﾁ ﾁ ﾓ16ﾓ16ﾓ1 A 4 A 2 ﾁ 6ﾓ16ﾓ16ﾓ1 A 4 A 3 : ( 6ﾓ16ﾓ16ﾓ1 A 4 M; 6ﾓ16ﾓ16ﾓ1 A 4 A 2 ; 6ﾓ16ﾓ16ﾓ1 A 4 A 3 ) = x 1 ( 6ﾓ16ﾓ16ﾓ1 A 4 A 1 ; 6ﾓ16ﾓ16ﾓ1 A 4 A 2 ; 6ﾓ16ﾓ16ﾓ1 A 4 A 3 ): 4 ﾆ i=1 x i 6ﾓ16ﾓ1 OAi ﾁ ﾁ x 1 = V MA 2 A 3 A 4 V A1 A 2 A 3 A 4. ｨ ﾁ02 ﾁ x 2, x 3, x ｨｰ OBCD, AOCD, ABOD 07 ABCO, O ﾁ ﾂ ﾁ0206, ﾁ , ﾁ ABCD ( ﾁ , O ﾁ ). ﾂ03- ﾁ , ﾁ ﾁ, ﾁ O, 0609 ﾁ ﾁ ﾁ, , ﾁ ﾁ ﾁ O ﾁ0203 ﾁ , BCD ﾁ O 1 BCD V AO1 CD, V ABO1 D 07 V ABCO ﾁ ﾁ ﾁ03 ﾁ , ﾁ ﾁ O 1, x 1 : x 2 : x 3 : x 4 = 6ﾓ1S 1 : S 2 : S 3 : S 4. ｨ ﾁ ﾁ0203 ﾁ A 1 A 2 A 3 A , 09 Q ﾁ A i A j a ij ﾁ A i S i ﾁ , 09 O i ﾁ0203 ﾁ , O k (x 1 ; x 2 ; x 2 ; x 4 ) ﾁ Q, 0803, ﾁ , ﾆ x i x j a 2 ij > O 1 i<j x 1 : x 2 : x 3 : x 4 = 6ﾓ1S 1 : S 2 : S 3 : S 4, ﾁ ﾁ ﾁ ﾁ ﾓ1S 1 S 2 a ﾓ1 S 1 S 3 a ﾓ1 S 1 S 4 a S 2 S 3 a S 2 S 4 a S 3 S 4 a 2 34 > 0: (21) O 2, O 3 07 O ﾁ Q, ﾁ ﾁ09, (21). ｨｮ ﾁ ﾁ ﾁ ﾁ , i 07 j S i S j a 2 ij ﾁ ﾁ ﾁ ｨｲ , > ﾁ ﾁ ﾁ , O i ﾁ ﾁ ｨ ﾁ ﾁ ﾁ ﾁ0203 ﾁ ﾁ0203 Q ( ﾁ03 ﾁ ﾁ ﾁ > <) , ﾁ O i Q, ﾁ Q

47 13ｨｪ ﾁ ﾂ ﾁ S 1 = S 2 = S 3 = S 4, a 12 = a 34, a 13 = a 24, a 14 = a ﾁ ﾁ ﾁ09 (21) 0609 ﾁ , ﾁ O Q. ｨ O i. ｨｲ ｨｦ ﾁ : ﾁ0203 ﾁ , ﾁ ﾁ ﾁ [10] s ﾁ ﾁ ﾁ ﾁ0703 ﾁ ﾁ ﾁ , , ﾁ , ﾁ s, 0403 ﾁ , 0609 ﾁ ﾁ ﾁ ｨｲ , s = ﾁ , , ﾁ , ｨｰ ﾁ , ﾁ ﾁ , ﾁ O ﾁ , (y + t)a + (z + x)b + (t + y)c + (x + z)d = (ya + xb) + (zb + yc) + (tc + zd) + (xd + ta) = = (x + y)a 1 + (y + z)b 1 + (z + t)c 1 + (t + x)d 1 ; (x + y) 6ﾓ16ﾓ1 OA 1 + (y + z) 6ﾓ16ﾓ1 OB 1 + (z + t) 6ﾓ16ﾓ1 OC 1 + (t + x) 6ﾓ16ﾓ1 OD 1 = 0: ｨｲ , ﾁ ﾁ ｨｲ , E = c((x + z)b; (x + z)d); F = c((y + t)a; (y + t)c), O = c(2(x + z)e; 2(y + t)f) ﾊ EF: ｨｲ ｨｮ03 ﾁ ﾁ ﾁ [1], c ｨｮ , 0908 ﾁ ﾁ ﾁ : ﾁ ﾁ , , ﾁ , ﾁ ﾁ ﾁ , , 02 ﾁ ABCD ﾁ a = DA, 6ﾓ1 b 6ﾓ16ﾓ1 = DB, c 6ﾓ16ﾓ1 = DC ﾁ ﾁ c ﾁ b, a ﾁ c, b ﾁ a, 6ﾓ1 AB ﾁ 6ﾓ1 AC = (b 6ﾓ1 a) ﾁ (b 6ﾓ1 a) , ﾁ ﾁ ( , ﾁ ﾁ03 ﾁ ), ﾁ ﾁ ﾁ ﾁ ﾁ ｨｦ ﾁ ﾁ09 ﾁ ﾁ , ﾁ , c ﾁ b + a ﾁ c + b ﾁ a + (b 6ﾓ1 a) ﾁ (b 6ﾓ1 a) = 0:

48 1346 ｨ. ｨ. ｨｹ ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ , ｨ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ( ﾁ, ﾁ ), ﾁ ﾂ , ﾁ ﾁ ﾁ , ﾁ , ﾁ , ﾁ ﾁ 0609 ﾁ Q ﾁ , V i ﾁ i ﾁ M i, n i ﾁ , i ﾁ ﾁ03 ﾁ , S i i s = ﾆ S i n i. ｨｰ ﾁ X ﾁ Q i ﾁ ﾁ ﾁ 6ﾓ16ﾓ16ﾓ1 XM i n i. ｨｮ ﾁ X 07 ﾁ , , ﾁ i , 0609 ﾁ S i 6ﾓ16ﾓ16ﾓ1 XM i n i V = ﾆ S i 6ﾓ16ﾓ16ﾓ1 XM i n i = ﾆ S i ( 6ﾓ16ﾓ1 XY + 6ﾓ16ﾓ1 Y M i ) n i = = 6ﾓ16ﾓ1 XY ﾆ S i n i + ﾆ S i 6ﾓ16ﾓ1 Y M i n i = 6ﾓ16ﾓ1 XY s + 3V; X 07 Y ﾁ ﾁ Q ﾓ16ﾓ1 XY s = s ﾙ 0, ﾁ X 07 Y ﾁ ﾁ 6ﾓ16ﾓ1 XY 07 s, ﾁ ﾁ ﾁ09 6ﾓ16ﾓ1 XY s = 0. ｨｲ ﾁ ﾁ [10] 07 [12] O ﾁ ｨｰ ﾁ AB, 09 K 07 M AK = AM, BK = BM ( , ﾁ , 0609 ﾁ ). ｨｲ , ABK 07 ABM 0609 ﾁ ( ) ﾁ ﾁ , ﾁ020403, , ﾁ , 0609 ﾁ0202. ｨｲ , ﾁ ﾁ , , ﾁ ﾁ , , ﾁ ﾁ ﾁ , 0609 ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ , 0308 ﾁ , 0803, , ﾁ , ﾁ ﾁ ﾁ ﾁ ｨｹ , ﾁ , ﾁ ﾁ ｨｮ , , ﾁ ﾁ, ﾁ , 0609 ﾁ0202. ｨｲ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ A 1 A 2, B 1 B 2 07 C 1 C , O, , , ｨｲ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ( ﾁ0907) ﾁ , 0609 ﾁ , 0609 ﾁ ﾁ ﾁ ﾁ , ｨｰ ,

49 , , ﾁ ﾁ (07 ﾁ ﾁ ﾁ ﾁ, ). ｨｬ [1] ｨｪ.09., ﾂ { ｨｪ.: , { [2] ｨｹ ﾁ ｨ.ｨ. ｨｪ ﾁ == ｨｪ ﾁ { { { ｨｮ. 59{67. [3] ｨｦ : 9{ ｨｪ.: , { [4] ｨｹ ﾁ ｨ.ｨ : ﾂ , { [5] ｨｹ ﾁ ｨ.ｨ. ｨｪ ﾁ ｨ : ｨｦ ｨ , { [6] ｨｹ ﾁ ｨ.ｨ ﾁ ﾁ. ｨｪ.: 05ｨｰｨ ｨｮｨ 0601, { [7] ｨｹ : ﾂ { 00. 2: ｨｮ , ﾁ ﾁ09. ｨｪ.: ｨｪ0006ｨｪ07, { [8] ｨｹ : ﾂ { 00. 3: ｨｪ.: ｨｪ0006ｨｪ07, { [9] ﾁ ﾂ. ﾂ. ｨｲ : ｨｪ.: , { [10] ﾁ ﾂ. ﾂ., ｨｦ.03. ｨｲ ｨｪ.: , { [11] ｨｪ ﾁ ｨ.04. ｨｹ ｨｪ.: ｨｪ0006ｨｪ07, { [12] ｨｪ ﾁ == 05 ﾁ { { { ｨｮ. 2{8. 47 ｨｹ ﾁ ｨ ｨ ﾁ0707, ｨ ﾁ ﾁ , [email protected]

50 13ｨｮ ﾁ ﾁ ﾁ ﾂ. ﾂ. ｨｦ ﾁ0003 ﾁ ｨｰ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｬ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｬ ﾂ , ﾁ ﾁ , , , ﾁ ﾁ [0; ﾞ) ﾁ , ﾂ ﾁ ﾁ ｨｬ ﾁ ﾁ ｨｹ ﾁ ﾁ : ｨｬ L () k (x) k ｨｰ L () k (x) = ex x d k k! dx k (xk+ e 6ﾓ1x ): (1) ﾁ (1) L () k (x) = k ﾆ i=0 C k6ﾓ1i k ﾁ ﾁ ｨｬ ﾁ0702 ﾒ ﾞ 0 (6ﾓ1x) i ; > 6ﾓ11: (2) i! (x) [ L () k (x)] 2 dx = 04( + 1)C k+ ; (x) = x e 6ﾓ1x : (3) f(x) ﾁ ｨｬ : a k = f(x) = ﾒ ﾞ n ﾆ k=0 a k L () k (x) (4) 0 (x)f(x)l () k (x)dx ﾒ ﾞ 0 (x) [ L () k (x)] 2 : (5) dx ﾁ f(x) ﾁ (x 1 ; :::; x n ) [0; ﾞ) ﾁ ﾁ ｨｨ , ﾁ , ﾁ ﾁ m k = 1 n 48 n ﾆ i=1 x k i (6)

51 ﾁ ﾁ ﾁ " 0706 ﾁ a k [1], [2]. ｨｰ ﾁ a k = ﾒ ﾞ f(x) = [(x)] 6ﾓ11 f(x) 0 (x)(x) 6ﾓ11 f(x)l () k (x)dx ﾒ ﾞ 0 (x) [ L () k (x)] 2 = dx ﾁ0701 ﾁ (7) L () k (x) 0703 (2) (3): ﾒ ﾞ 0 f(x)l () k (x)dx ﾒ ﾞ 0 (x) [ L () k (x)] 2 : (7) dx k ﾆ Ck6ﾓ1i k+ i=0 (i)6ﾓ11 ﾒ ﾞ 0 x k k ﾆ i f(x)dx Ck6ﾓ1i k+ m i i! (6ﾓ11)i a k = 04( + 1)Ck+ i=0 = 04( + 1)Ck+ : (8) m i ﾁ a k ﾁ , m k = ﾒ ﾞ 0 x k f(x)dx m k = 1 n , f(x), , : f(x) = (x) e 6ﾓ1x n ﾆ L () k a k (; ) = k=0 (x) = k ﾆ i=0 1 04( + 1)C k+ n ﾆ i=0 x k i : a k (; )L () k (x); (9) C k6ﾓ1i k+ k ﾆ i=0 (6ﾓ1x) i ; (10) i! C k6ﾓ1i k+ (6ﾓ1) i m i : (11) i! ﾂ ﾁ m i , m i ﾁ m i , ﾁ ( ﾁ, ﾁ ﾁ03 ﾁ,... ) ﾁ ﾁ ﾁ 07 ｨｦ03 (9) ﾁ , f(x) , 0309 ﾁ ﾁ a k k = n ﾁ n ﾁ ﾁ ﾁ , , ﾁ ﾂ ﾁ, ﾁ f(x) ﾁ0203, (3): Y n = ﾒ ﾞ 0 (x) [ f(x) 6ﾓ1 n ﾆ k=0 a k (; )L () k (x) ] 2 dx = ﾞ ﾆ k=n ﾁ ﾁ0708 a 2 k(; )04( + 1)C k+: = = 0:

52 1350 ﾂ. ﾂ. ｨｦ ﾁ0003 ﾁ ﾁ n = n ﾁ a n0 +3, a n0 +4, ﾁ a n0 +1, a n0 +2, ﾁ (12) ﾁ ﾁ a 2 n0 +1(; )Cn a 2 n 0 +2(; )Cn ] = 0; [ a 2 n0 +1(; ) + a 2 n 0 +2(; ) ] ﾂ (13) (+1), 0203 ﾁ ｨｰ (13) 05 ﾁ ﾁ0705 (13) a n0 +1(; ) = 0; a n0 +2(; ) = 0 (14) (4) , , ﾁ m n0 +1, m n0 +2, ﾁ a k ﾁ ﾁ ﾁ ﾁ = m 1 ; b i() = m 1 ( + 1) i m i 1( + 1):::( + i) ; i ﾝ 2; b 0 = b 1 = 1 (15) ﾂ (14) n 0 +1 ﾆ i=0 Ci n 0 +1(6ﾓ1) i b i () = 0 n 0 +2 ﾆ i=0 Ci n 0 +2(6ﾓ1) i b i () = 0 ﾂ (16) (+1) ﾁ , (16) 0203 ﾁ , n , ﾁ (16) n 0 = 0. ｨｦ ﾁ { 1 6ﾓ1 = 0 1 6ﾓ b 2 = = 1, b 2 = 1. ﾂ0303 ﾁ (15) m 0 = 1 m 2 6ﾓ1 m 2 ; 0 = 2m2 1 6ﾓ1 m 2 1 m 2 6ﾓ1 m 2 : (17) 1 ｨｰ ﾁ (9) n 0 ﾝ 1. ｨｰ (16) ﾁ ; ﾁ a 1, a 2, ﾁ ; ﾁ n ｨｦ (17) n 0 = ﾁ m 3, m 4, a 3 ( 0 ; 0 ), a 4 ( 0 ; 0 ), (16) ( 0 = 1): a 3 = 1 04( 0 + 1) (1 6ﾓ1 b 3( 0 ; 0 )); a 4 = 1 04( 0 + 1) (3 + 4b 3( 0 ; 0 ) + b 4 ( 0 ; 0 )); ::: b 3 ( 0 ; 0 ); b 4 ( 0 ; 0 ) (15). ﾂ , ﾁ ﾁ ﾁ ﾁ (9), ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ0301. ｨｮ ﾁ a k = ﾁ ﾁ ﾁ ﾁ (16)

53 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ (9) ﾁ ﾁ ﾁ ﾁ ﾁ (x 1,...,x 2 ) m k ﾁ ﾁ ﾓ5 07 6ﾓ5, m i = (6ﾓ5 + i)! 6ﾓ5!( 6ﾓ5 ) i ; i ﾝ 1; m 0 = 1: ﾁ ﾁ m i ﾁ (15): = ( ) 6ﾓ5 6ﾓ11 ; bi = 6ﾓ ( 6ﾓ5 + 1) i ( ( 6ﾓ5 + 1):::( 6ﾓ5 + i) = 1; b 2 = 1; ﾓ5 = ﾓ5 = 0 : ( 0 + 1) i ( 0 + 1):::( 0 + i) ) 6ﾓ11 : ｨｹ ﾁ i > 2, b i = 1, i = 3; n ﾁ a k k ﾝ ﾁ a k = 04( 0 + 1) (1 6ﾓ1 )k = 0; = ﾁ ｨｰ ﾁ ﾁ f(x) = 0 ( 0 x) e 6ﾓ1 0x a 0 L 0 (x) = 0( 0 x) 0e 6ﾓ1 0x 0! a 0 = 1 04( 0 +1), L() 0 (x) = f(x) [0; ﾞ) ﾁ ｨｨ (9) f(x) ﾁ ﾁ ﾁ f(x) = ﾁ03. ｨｦ n ﾆ k=0 p k f k (; ; x); n ﾆ k=0 p k = 1; p k ﾝ 0: L () k (9) ﾁ0702 (x) = k ﾆ i=0 Ck+ k6ﾓ1i (6ﾓ1x) i i! = k ﾆ i=0 (x) i ( + i)! Ck6ﾓ1i ( + i)! k+ (6ﾓ11) i! i : f(x) = (x) e 6ﾓ1x ﾆ n a k (; ) k=0 ﾂ (18) ﾁ : f(x) = e 6ﾓ1x k ﾆ i=0 (x) +i ( + i)! n ﾆ k=i k ﾆ i=0 (x) i ( + i)! Ck6ﾓ1i ( + i)! k+ (6ﾓ11) i! i : (18) a k (; )(6ﾓ11) i C k+ ﾂ (19) e 6ﾓ1x ﾁ f i (x) = (x)+i e 6ﾓ1x ( + i)! ( + i)! : (19) i! ﾓ i 07 :

54 1352 ﾂ. ﾂ. ｨｦ ﾁ0003 ﾁ ﾂ ﾁ (19) ﾁ p i, p i = n ﾆ k=i a k (; )(6ﾓ11) i C k+ ( + i)! ; f(x) = i! ﾆ , n p ﾆ i = 1. ｨｦ n p ﾆ i = n a k(; )(6ﾓ11) i C k6ﾓ1i i=0 i=0 i=0 k=i 0603 ﾁ : n ﾆ i=0 p i = n ﾆ i=0 n ﾆ a k (; ) k ﾆ C k6ﾓ1i i=0 ｨｰ ﾁ ﾁ ﾁ : n ﾆ i=0 k+ (+i)! k+ (6ﾓ11)i ( + i)! : i! p i f i (x): i!. ｨｦ k ﾆ i=0 Ck+ k6ﾓ1i (6ﾓ11)i ( + i)! = i! k ﾆ i=0 (k + )!( + i)!k! (k 6ﾓ1 i)!( + i)!i!k! (6ﾓ11)i = k ﾆ i=0 k!(6ﾓ11) i (k + )! i!(k 6ﾓ1 i)!k! k = 1; n ﾁ a 0 (; ) k = 0. = (k + )! k! k ﾆ i=0 C i k(6ﾓ11) i : n ﾆ i=0 p i =!a 0 (; ) = 1; a 0 (; ) = 1! : , p i < 0, ﾁ f(x) , ﾁ , f(x) , ﾁ , f(x) ﾝ 0. ｨｮ , ﾒ ﾞ 0 f(x)dx = n ﾁ , ﾒ ﾞ 0 f(x)dx = ﾒ ﾞ 0 n ﾆ i=1 p i f i (x)dx = n ﾆ i=1 ﾒ ﾞ 0 p i f(x)dx = n ﾆ ﾒ ﾞ p i i=1 0 ﾂ : 1) ｨｹ ( ) f i (x) ﾁ f(x)dx = n ﾆ i=1 p i = 1: x i max = + i ; i = 0; n! ｨｨ ﾁ ) ｨｲ f k (x) ﾁ x k max ﾁ ) ｨｨ ﾁ030708" f n (x) ﾁ (x n max ; ﾞ) 05 ﾁ ﾁ ﾁ ﾁ ｨｬ ｨｦ ﾁ0003 ﾁ ﾂ. ﾂ ﾁ ﾁ ｨｬ ﾁ // ｨｦ03 ﾁ ｨ 06 ｨｮｨｮｨｮｨｰ. - ｨｪ.: ｨｦ ﾁ0003 ﾁ ﾂ. ﾂ ﾁ ﾁ0906 ﾁ , ｨｬ / ｨｮ ﾁ. - ｨｪ.: ｨｪ , ｨｦ ﾁ0003 ﾁ ﾂ ﾂ ﾁ0707, ｨｪ ﾁ ﾁ ﾁ ｨｪ.ｨ ﾁ09, [email protected]

55 13ｨｰ ﾁ ﾁ09, ｨｮ. ｨ ﾁ ｨ ﾁ ﾁ090808, ﾁ ﾁ ﾁ ﾁ ( , ﾁ ﾁ ), ﾁ0205 ﾁ ﾁ ﾁ , , ( ﾁ , , ) ﾁ ﾁ ﾁ ﾁ , , ﾂ ( , ) ﾁ ﾁ ﾁ ﾁ ｨｮ ﾁ ﾁ ﾁ n", n ｨｦ ﾁ , 0609 ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ090801, w = u + v, , w = uv. ｨｹ ﾁ ﾁ ﾁ ﾁ ": log(uv) = log u + log v. (ｨｦ ﾁ ﾁ , ﾁ ) ｨ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ [Lyub]. ﾂ ﾁ ( ﾁ ﾁ ﾁ ) ﾁ ﾁ ﾁ , w = u + v, ( ). ﾂ , w = (u + v)=(1 + uv=c 2 ), 0403 ｨｬ ｨｹ ( ﾁ ). ﾂ ﾁ , ﾁ ﾁ k(v) = (c + v)=(c 6ﾓ1 v) k(w) = k(u)k(v) 0509 ﾁ07 ﾁ ｨｬ ｨｹ ﾂ k(v) , , v , c" ﾁ (07 ﾁ ﾁ !). ﾂ k(v) ﾁ ﾁ ( ") ﾁ030809, ﾁ v ﾁ0303 ﾁ ﾁ0303 ﾁ ﾁ ﾁ , 0609 ﾁ k(v) (ｨｹ ﾁ ﾁ ﾁ ﾁ070209, , ﾁ ﾁ c.) ﾁ k(v), log k(w) = log k(u) + log k(v), ﾁ , w " u 07 v 0403 ｨｬ ｨｹ ﾂ log k(v) ﾁ ( ) " 02 ﾁ ( , " " ﾁ ) ｨｬ ｨｹ tg(a + b) = (tg a + tg b)=(1 6ﾓ1 tg a tg b): ｨｹ ﾁ020308, k(a) = (1 + i tg a)=(1 6ﾓ1 i tg a) ﾁ Ln ﾁ k(a) [Lyub], ﾁ , Ln k(a) = 2ia. ｨｹ ｨｹ Exp(ia) = cos a+i sin a. (ｨｲ Exp , Ln [Lyub] ﾁ ﾁ ﾁ03: Exp(a+b) = Exp(a) Exp(b). ﾂ Ln, Exp a ﾁ a ｨｦ ｨｹ , ﾁ , cos 07 sin.) , k(a) = Exp(2ia), ｨｹ k(a) ﾁ , cos 2a = (16ﾓ1tg 2 a)=(1+tg 2 a) 07 sin 2a = 2 tg a=(1+tg 2 a) ( ﾁ ") , tg a = sin 2a=(1 + cos 2a). 53

56 ﾁ09, ｨｮ. ｨ ﾁ ｨｮ , ( [040904]) ﾁ ﾁ x = cos 2a, y = sin 2a ﾁ t = tg a. ｨ , ﾁ a, b, c ﾁ ( ﾁ , , ): a = n 2 6ﾓ1 m 2 ; b = 2nm; c = n 2 + m 2 : , ﾁ n = 2, m = 1, ", ﾁ : a = 3, b = 4, c = ﾁ ﾁ ﾁ x 2 6ﾓ1 y 2 = ﾁ ﾁ : Cosh a := (Exp a + Exp(6ﾓ1a))=2; Sinh a := (Exp a 6ﾓ1 Exp(6ﾓ1a)=2; tanh a := Sinh a=cosh a: ﾂ0303 ﾁ ｨｬ ｨｹ , k(v) = (1+tanh a)=(16ﾓ1tanh a), tanh a = v=c, a = (ln k(v))= ", ﾁ ﾁ ( ) ﾁ ｨｹ , ﾁ : Cosh(ia) = cos a; Sinh(ia) = i sin a; tanh(ia) = i tg a: ｨｹ ﾁ , x 2 6ﾓ1 y 2 = ﾁ ﾁ y iy. ｨｮ ﾁ ｨｹ , , ﾁ ﾁ ﾁ , " ﾁ a + be, e 2 = 1 ([0905]). (01 ﾁ , ﾁ : ﾁ ﾁ ﾁ ( ) ﾁ , (1 + e), (1 6ﾓ1 e).) Exp(ea) = Cosh a + e Sinh a ﾁ ｨｹ (ｨｪ " ea. ｨｮ ﾁ ﾁ ﾁ ﾁ ) ﾁ ﾁ ﾁ " ﾁ ( ﾁ03 02 ﾁ " ﾁ ﾁ 02 ﾁ !) k(v) = (c + ev)=(c 6ﾓ1 ev), e " , e 2 = 1, ﾁ ﾁ03 k(w) = k(u)k(v) ﾁ ﾁ ﾁ ( ) k(v) = exp(2ea), a ".

57 13ｨｰ ﾁ ｨｬ [040904] ﾁ0707 ｨｦ.ｨｰ ﾁ ｨｪ.: , [Lyub] ｨｬ ﾂ.ｨ ﾁ ｨｪ.: ｨ , [0905] ｨｦ.ｨｪ ﾁ ｨｪ.: , ﾁ ﾁ0209, ｨｦ ｨ. ｨ. ｨｨ ﾁ ｨｰｨ 06, [email protected] ﾁ ｨｮ ｨ ﾁ0707, ﾁ ｨｦ ｨ. ｨ. ｨｨ ﾁ ｨｰｨ 06, [email protected]

58 13ｨｮ ﾁ : ﾁ ? 05. ｨｬ ﾂ , , ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ｨｹ ﾁ ﾁ ﾁ ﾁ; , ﾁ , ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ , ﾁ ﾁ ﾁ ﾁ ( ﾁ ﾁ ) 07, ﾁ ﾁ , ﾁ , ﾁ ﾁ ( ﾂ ", , ) 1, ﾁ090308, ﾁ 0803 ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ ｨｹ ﾁ0203 ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ｨｪ ﾁ ｨｰ03 ﾁ ｨｲ ﾁ ｨｦ ﾁ ﾁ 05 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0301. ﾂ , ﾁ ﾁ ﾁ , ﾁ , ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾂ ﾁ ﾁ0703 ﾁ , , , ﾁ ﾁ ﾂ ﾁ " ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ 07 ﾁ vofem.ru

59 ﾁ ? ﾁ , ﾁ , ﾁ090308, , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ ?"; ﾁ ﾁ , , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ 0308 ﾁ , ﾁ , 09 ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0705 ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ , ﾁ , , ﾁ ﾁ ﾁ ｨｰ , ﾁ ｨ, ｨ ﾂ, ｨ ﾂ , ｨ ﾂ, ﾁ09 ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ; , ﾁ ﾁ , , , ﾁ ﾁ ﾁ ﾁ0202 ﾁ ﾁ , priori ﾁ ﾁ ﾁ0209, ﾁ ﾁ ﾁ , ﾁ , ﾁ , ﾁ0308 ﾁ , ﾁ ; 0203 ﾁ ﾁ , , ﾁ , ﾁ ; ﾁ , ﾁ030007, ﾁ ﾁ ｨｲ , ﾁ ﾁ , ﾁ , ﾁ, ﾁ ﾁ ﾁ ｨｮ0701 ﾁ0300 ｨ ﾂ ﾁ ﾂ ﾁ0206, ; ｨ ﾂ ﾁ090202; ｨｮ ﾂ ﾁ ｨ ﾂ, ｨｮ 0703 ﾁ ｨ, 09 D 07 ﾂ: ﾂ03- ﾁ , ﾁ ｨ ﾂ, ﾁ0708: ﾁ03 ﾁ ﾁ ﾁ09 ｨ ﾂ = ｨｮ ﾂ , ｨ ﾂ 07 ｨｮD ﾁ ﾁ ﾁ , ｨ ﾂ; , ﾁ , , , ﾁ0301, ﾁ ", ; ﾁ03 ﾁ , 0403 ﾁ , ﾁ , ﾁ ｨ ﾂ ﾁ , 07 ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ030605, ｨ ﾂ = ｨｮD, ﾁ ｨ ﾂ

60 ｨｬ ｨｮD , ﾁ ﾁ ﾁ ; ﾁ , ｨｮ (Hamilton), , ﾁ ﾁ ﾁ ; (Grassmann) ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨ ﾂ = ｨｮ ﾂ, ﾂ 07 ｨ, ﾁ , 0609 ﾁ D 07 ｨｮ, , = 7 6ﾓ1 4 = 20 6ﾓ1 17 = (12 + 2i) 6ﾓ1 (9 + 2i): ﾁ ﾁ " ﾁ ﾁ ﾁ , ﾁ , , ﾁ : 0609 ﾁ ﾁ ﾁ ﾁ" ﾁ ﾁ07 ﾁ ﾁ ﾁ", ﾁ 07, ﾁ ﾁ ﾁ ", ﾁ ﾁ. ｨｹ , ﾁ ﾁ ﾁ ﾁ ｨｦ , ﾁ ﾁ , ﾁ , , 02 ﾁ ﾁ0708 ( ), ﾁ , ﾁ ( ﾁ ﾁ03; ), 09 ﾁ ﾁ ( ). ｨｦ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ; , , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ03 ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ ; ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ , , ﾁ ﾁ ﾁ ﾁ , , ﾁ , ﾁ ﾁ ﾁ ﾁ , , ﾁ ; ﾁ , ﾁ ﾁ , ﾁ ｨｹ ﾁ ﾁ090801, ﾁ09 ﾁ , ﾁ ﾁ , ﾁ , , , ; ﾁ ﾁ ; ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ｨｮ. Bourali-Forti et R. Marcolongo, Elements de calcul vectoriel". ｨｰ ｨｦ ﾁ ｨｰ0302.

61 ﾁ ? 59 ﾂ03 ﾁ ﾁ ﾁ0703 ﾁ , ﾁ , , ﾁ 0309 ﾁ , ﾁ ﾁ ﾁ , ﾁ , , ﾁ , ﾁ ﾁ , ﾁ , ﾁ ﾁ0906 ﾁ ﾁ , , ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , , ﾁ , ﾁ , , ﾁ , ﾁ , , , ﾁ ﾁ , ﾁ ﾁ ﾁ070209, 0803, ﾁ , ﾁ ﾁ, ﾁ ﾁ02 ﾁ ﾁ, , ﾂ , ﾁ ﾁ ﾁ ﾁ, ﾁ (07, ﾁ , ) ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ; , ﾁ

62 13ｨｦ ﾁ ｨｦ. 05. ｨ ﾁ ﾁ03: ｨｮ 縺 ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ ｨｦ. 05. ｨ ﾁ ｨｦ ﾁ09 ｨｮ , 縺 ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ 07 ﾁ ( ) ﾁ ﾁ ｨｦ ﾁ ｨ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ { ﾁ ﾁ ﾁ03 ﾁ ﾁ , 0209 ﾁ , ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ 縺 ﾁ ﾁ ﾁ ﾁ ｨｰ ﾁ ｨ (ｨ 0806) ｨｮｨｮｨｮｨｰ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ｨｪ0708ｨｦ ｨｦ ﾁ ｨ ﾁ09 ( ) ｨ ﾁ ｨｦ ﾁ ( { ) ｨｦ. 05. ｨ ﾁ ﾁ ﾁ , , ｨ 0806 ｨｮｨｮｨｮｨｰ, ｨｲ

63 ｨｦ. 05. ｨ ﾁ ﾁ ｨｮｨｮｨｮｨｰ ﾁ ｨｮｨｮｨｮｨｰ, 0909 ﾁ ﾁ ﾁ : ｨｬ , ﾁ , ﾁ03 ﾁ { ﾁ, , ﾂ. ｨｦ. ｨｬ , , ｨｬ. ｨｹ , ｨｦ (0701. ｨ ﾁ ﾁ [16, c. 228]). ﾂ ｨｦ ﾁ ｨ ﾁ09: ﾁ ｨｦ ﾁ ｨ ﾁ ｨｪ0702 ﾁ ｨｮｨｮｨｮｨｰ, ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ｨｰｨｮ03ｨｮｨｰ, ﾁ03 ﾁ0900 ﾁ ﾁ ﾁ09 ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ09 罟ｲ , ﾁ ﾁ ﾁ ｨｰｨｮ03ｨｮｨｰ , ｨｦ ﾁ ﾂｨ 05 ｨｮｨｮｨｮｨｰ ｨｦ. 05. ｨ ﾁ 罟ｪ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 罟ｪ ﾁ 縺 ﾁ ｨｦ ﾁ ﾁ ﾁ , 0203 ﾁ ﾁ , ﾁ ﾁ, ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ [19]. ｨｦ ﾁ ﾁ , ﾁ ｨｦ. 05. ｨ ﾁ : , 0403 ﾁ ﾁ, ﾁ ﾁ. ﾂ ﾁ ﾁ ﾁ ﾁ , ﾂ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 縺0603 ﾁ ｨｦ ﾁ ﾁ 07 ﾁ ﾁ (0701. [16, ]), , ﾁ ﾁ0005 ﾁ ﾁ 0803 ﾁ , ﾁ0902 ﾁ 罟ｪ ﾁ (1918, 6 5 2) ﾁ ｨｦ. 05. ｨ ﾁ09, , ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ｨｮｨｮｨｮｨｰ ﾁ ﾁ 07 ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾂ , , ﾁ ﾁ , ｨ ｨ , ﾁ ﾁ , ﾁ ﾁ ｨ ﾁ ﾁ ﾁ ｨｦ ﾁ ｨ ﾁ ﾁ ﾁ , ﾁ ﾁ I{III ﾁ ﾁ ﾁ 罟ｪ ﾁ (1967, , 2) ﾁ ﾁ0301 ｨｦ ﾁ ﾁ0309, , ﾁ ﾁ ﾁ, ﾁ, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ ﾁ02 07 ｨｪ ﾁ , ﾁ ﾁ ﾁ, ﾁ 0403 ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ ｨｮｨｮｨｮｨｰ (ｨｬ , ｨ , ﾂ , ﾁ09, ｨｮ ﾁ09, , , , ｨｰ070507, ﾂ , ｨｮ ﾁ ﾁ070909, ｨｦ , ﾁ09, ) ｨｦ. 05. ｨ ﾁ ｨ.ｨｪ. ﾁ : 縺0003, ﾁ ｨｦ ﾁ , ,

64 ｨｦ ﾁ , , ( , ｨｦ ), !1 7 [10, ] ｨｦ. 05. ｨ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ IV{V ﾁ ﾁ ｨｮｨｮｨｮｨｰ, ﾁ ﾁ ﾁ ﾁ. ｨｹ , ﾁ , ﾁ ﾁ ﾁ0209, , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ , ﾂｨｪ05... ｨｦ ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ03 ﾁ ﾁ ( ﾂ ﾁ09, ﾁ), (ｨｮ , ﾂ ), ( ), ﾁ XV ｨｪ (ｨｪ ﾁ09, ) ﾁ ﾁ 縺0603 ﾁ ﾁ ﾁ , ﾁ ﾁ0707 ﾁ ﾁ ( ﾁ [8, ]) : 縺0203 ﾁ ( ) 縺0703 ﾁ ( ) 縺 ﾁ ( ) ﾁ 縺 ﾁ ﾁ ﾁ ﾁ ｨｰ ｨｮ ﾁ 08 ﾁ ﾁ : ｨｪｨｪ ｨｪ ｨｪ. ﾂ. ｨｬ ﾁ ﾁ ｨ 0806 ｨｮｨｮｨｮｨｰ { ﾁ 罟ｰ0903 ﾁ ﾁ ﾁ 縺 ﾁ ﾁ ﾁ ｨｦｨｦ ｨｮ07ｨｪ07 ｨ 0806 ｨｮｨｮｨｮｨｰ ｨｦ. 05. ｨ ﾁ , ﾁ ｨ ﾁ , ﾁ ( {7): 6ｦ ｨ 0806 ｨｮｨｮｨｮｨｰ, , ｨｦ06 ( ﾁ ｨｲ ﾁ09) ｨｦ ﾁ0902 ｨｮ ﾁ ﾁ ﾁ (1975{1981), 6ｦ ｨ 0806 ｨｮｨｮｨｮｨｰ / ｨｰｨ 07, , ｨｪ ﾁ ﾁ ｨｦ ﾁ ﾁ0003 ﾁ0707 ﾂ (1981{1984), 6ｦ1 ｨｲ ｨｰ03, , ｨｪ (ｨｪ ﾁ ﾁ ﾁ ) ﾂ ﾁ0707 ｨｪ ﾁ (1981{2009), 6ｦ , ｨｰ (ｨｰ ﾁ ﾁ), ﾁ ｨｪ ( ﾁ090909, ｨｪ ) ｨｬ03 ﾁ ﾂ ﾁ0707 ｨｮ (1981{2000), 6ｦ1 ｨｲ ｨｰ03, ｨｰｨ 07, , ｨｪ ｨｬ09 ﾁ0603 ﾁ0707 ｨｬ (1992{2006). ﾂ ﾂ. ｨｪ ﾁ ( ﾁ 07 ﾁ ) ﾁ ﾁ ｨｪ07 ｨｪ0401 ( ﾁ ｨｪ ﾁ ﾁ ﾁ ｨｪ. ﾂ. ｨｬ ﾁ09) ｨｦ ﾁ ﾁ ﾁ0308, 09 ﾁ ｨｦ : ﾁ ﾁ ｨｪ ｨｪ. ﾂ. ｨｬ ﾁ09, ﾁ ｨ 0806 ｨｮｨｮｨｮｨｰ / ｨｰｨ 07.

65 ｨｦ. 05. ｨ ﾁ ﾁ ﾁ ﾁ ｨｦ ﾁ0902 ｨｮ ﾁ0707 (1916{ ) ﾂ ｨｦ ﾁ ﾁ0003 ﾁ0707 ( { ) ﾁ ﾁ ﾁ0205 ﾁ ﾁ ﾁ , ﾁ ｨｰ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ( ﾁ [16, c. 117{124]): 1) 縺 ﾁ ﾁ , ﾁ ; 2) 罟ｰ ﾁ ﾁ ﾁ ﾁ ; 3) 罟ｦ ﾁ ; 4) 縺 ﾁ 07 ﾁ ﾁ ﾁ ; 5) 縺 ﾁ ; 6) 縺 ﾁ ; 7) 縺 ﾁ , ; 8) 罟ｦ ﾁ ﾁ ; 9) 罟ｰ ﾁ ; 10) 罟ｰ ﾁ ﾁ, ﾁ ﾁ ;

66 ｨｦ ﾁ ｨｪ ﾁ ﾂ ﾁ0707 ( { ) ｨｮ ｨｬ03 ﾁ ﾂ ﾁ0707 ( { ) ｨｬ ｨｬ09 ﾁ0603 ﾁ0707 ( { ) ﾁ ｨｦ ﾁ ﾁ0209 ( ) 11) 罟ｦ ﾁ ﾁ ﾁ ; 12) 縺 ﾁ ; 13) 縺 ﾁ ;

67 ｨｦ. 05. ｨ ﾁ ﾁ ) 縺 ﾁ ﾁ ﾁ ﾁ0105, ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ, ; 15) 縺 ﾁ ( ﾁ )1 7; 16) 縺 ﾁ ; 17) 縺 ﾁ , ﾁ ﾁ ; 18) 縺 ﾁ ; 19) 縺 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 07 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ ﾁ , ﾁ ﾁ07 ﾁ ﾁ0203 ﾁ ﾁ ﾁ ﾁ ｨｪ0708ｨｦ , ｨｦ ﾁ ｨ ﾁ09 [12] ( ) ｨ ﾁ ｨｦ ﾁ0707 ( { ) ｨｮ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ : ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾂ ﾂ ﾁ ｨｦ. 05. ｨ ﾁ09, ﾁ ﾁ0301 ﾁ , ﾁ ﾁ ﾁ ｨ ( 縺01 ﾁ ﾁ ﾁ ﾁ ﾂ ), ( ).

68 ｨｦ ﾁ09 ｨｮ ( ﾁ ) ﾂ , ﾁ , ﾁ ﾂ ﾁ0209 ( { ) ｨｨ ﾂ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ｨｰ , ﾁ , ﾁ ﾁ ﾁ , ﾁ ｨｲ ﾁ. ｨｮ0901 ｨｦ ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ , ﾁ ﾁ 0703 ﾁ ﾁ ﾁ ﾁ , ﾁ , , , , ｨｦ. 06. ｨ ﾁ, ｨｦ. ｨｮ ﾁ ﾁ, ﾂ. ｨ ﾁ, ﾂ. ｨ , ﾂ. ｨ ﾁ0707, 01. ｨｦ. ｨｦ ﾁ, ｨ. ｨｪ , ﾂ. ｨ , 04. ｨｬ. ｨｬ , 07. ﾂ. ｨｪ ﾁ, ｨ. ｨｦ. ｨｪ ﾁ0707, ｨｪ ﾁ09, 06. ｨ. ｨｪ , ﾂ. ｨ , 04. ｨｦ. ｨｮ ﾁ, ｨｲ. ｨ. ｨｮ , ｨ. ｨ. ｨｮ , 08. ﾂ. ｨｮ ﾁ, , ｨ. ｨｦ ﾁ, ﾂ. ﾂ , ｨｰ. ｨｮ ﾁ, ﾁ , ｨｦ , 08. ｨ ﾁ ﾁ ﾁ, ｨｦ. 05. ｨ ﾁ0201, ﾁ ﾁ ﾁ , ﾁ 07 ﾁ , ﾁ ﾁ (ｨｰ ｨｰ ), ( ), ｨ. ｨｦ ﾁ09 (ｨｰ ), ｨｬ (ｨｰ ), (ｨ ﾁ0705), 08. ｨｦ ﾁ ﾁ ( ), ｨｮ ﾁ ( ), ｨｮ. ｨｪ ( ), ﾂ. ｨｪ ( ﾁ ), ( ﾂ ), ( ), ｨ ( ﾂ ), ( ) ｨｮ , ﾁ ﾁ ｨｲ ﾁ ﾁ ﾁ ｨｪ0708ｨｦ (1959{1981), ｨｪ ﾁ ﾁ ﾁ03 ﾁ (1981{ 1992) , ﾁ ﾁ ﾁ ｨｪ0708ｨｦ, ﾁ ﾁ , ﾁ { ﾁ ﾁ ﾁ ｨｮ ﾁ09 ﾂ ｨ ﾁ ｨｦ. 05. ｨ ﾁ09. ｨｹ ﾁ ﾁ , ﾁ07 ﾁ ﾁ ｨｦ. 05. ｨ ﾁ09: ﾁ 0403 ﾁ , , ｨｦ ﾁ ﾁ ﾁ , ﾁ ﾁ0301, ﾁ

69 ｨｦ. 05. ｨ ﾁ ﾁ03 67 縺 ﾁ ﾁ ｨｪ0708ｨｦ ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ , ﾁ ﾁ ﾁ ﾁ, ﾁ , ﾁ ( ﾁ), : 縺 , ﾁ , ﾁ ﾁ ﾁ , ｨｦ. ｨｬ ﾁ ﾂ ﾁ ｨｪ0708ｨｦ ( , , , , ﾁ , ﾁ ﾁ ｨｮｨｮｨｮｨｰ) ﾁ 縺 ｨｹ 縺 ｨ ﾁ ﾁ , ﾁ ﾁ03 ﾁ : 縺 ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ ﾂ. ｨ. ｨｮ ﾁ09, ﾁ ﾁ ｨ ﾁ09, 縺 ﾂ ｨｪ0708ｨｦ ﾂ. ｨ. ｨｮ ﾁ ﾁ ﾁ ( ) ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ : ｨｦ. 05. ｨ ﾁ09 (I{III ) 07 ｨ ﾁ09 (IV{XI ). ｨｮ ｨ ﾁ 07 ｨｦ. 05. ｨ ﾁ " ﾁ ﾂ. ｨ. ｨｮ ﾁ0201 ﾁ ﾁ [17] ( ) ﾂ. ｨ. ｨｮ ﾁ , 07 ﾁ ﾁ ﾁ ﾁ :

70 ｨｦ ﾁ ｨ ﾁ ﾁ ｨｪ0708ｨｦ (ｨｪ040701) ( ｨｮ ﾁ09 ﾂ.ｨ.) ｨｦ. 05. ｨ ﾁ09 07 ｨ ﾁ09 ｨ ﾁ ｨｦ ﾁ ｨ ﾁ ﾁ ! ﾁ , ﾁ ｨｰ ﾁ ﾁ ﾁ0703 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ ｨｦ ﾂ ﾁ09 ｨｰ ｨ ﾁ ｨｦ ﾁ ｨ ﾁ ﾁ

71 ｨｦ. 05. ｨ ﾁ ﾁ ! ﾁ ﾁ ｨｲ ﾁ ﾁ ﾁ ﾁ0309. ｨｪ ﾁ ﾁ: , , ｨｦ ﾁ ﾁ ﾁ , , ﾁ, ﾁ ｨ ﾁ ﾁ ｨｦ ﾁ ｨ ﾁ ﾁ090801? 0609 ﾁ , ﾁ , ﾁ ﾁ , ﾁ ﾁ030909, , ﾁ ﾁ ! ｨｮ , ﾁ ﾁ , ﾁ ﾁ [9; 16] ｨｦ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , 縺 , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 07 ﾁ ｨｦ ﾁ ﾁ ﾁ ｨｪ0708ｨｦ [1], 02 ﾁ ﾁ03 ﾁ ﾁ , ﾁ ﾁ. ｨｮ ﾂ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 2003{ ﾁ ﾁ ﾁ ﾁ ﾁ ﾂ. ｨ. ｨｮ ﾁ, ﾁ ﾁ , ﾁ , ﾁ030809, ﾂ ﾁ0209 ﾁ [2]. ﾂ ﾁ , ﾂ ﾂ. ｨ. ｨｮ ﾁ0201, ﾁ ﾁ ﾂ ｨ ﾁ ﾁ ﾂ ﾁ [2, 07. I, c. II]: ﾂ ｨ ﾁ ﾁ03, ｨｬ ｨｦ 0203 ﾁ ｨ ﾁ { ﾁ ﾁ ? ｨｦ ﾁ : ﾁ ｨ ｨ ﾁ , , ﾁ ﾂ ﾁ0208 ﾁ0309, ｨ 02, ﾁ ﾁ03 ﾁ , ｨ ﾁ ﾁ ｨｦ ﾁ ｨ ｨｦ 縺 ｨｦ. 05. ｨ ﾁ ﾁ ｨｦ ﾁ

72 ｨｦ ﾁ ﾁ , ﾁ ｨｦ. 05. ｨ ﾁ ﾂ ﾁ ｨｦ. 05. ｨ ﾁ ﾁ ﾁ 07 ﾁ ﾂ. ｨ. ｨｮ ﾁ09 罍ﾂ ﾁ , ｨｦ. 05. ｨ ﾁ03, ﾁ , , ﾁ [16] ( ), ﾁ ｨｰ ﾁ ﾂ.ｨ. ｨｮ ﾁ09 罍ﾂ ﾁ , ｨｹ ﾁ ﾁ ﾁ, ﾂ ﾁ ﾁ ﾁ ｨｦ. 05. ｨ ﾁ09 ( ), ( ), ( ), ﾁ ﾁ ｨｦ ﾁ ( ). ｨｦ { ﾁ ﾁ ﾁ ﾂ. ｨ. ｨｮ ﾁ ﾁ03 ﾂ ﾁ ﾁ ﾂ ﾁ (c. 7): ﾁ ｨｰ ! ｨ ﾁ ｨｦ ﾁ ﾁ: ﾁ ﾁ

73 ｨｦ. 05. ｨ ﾁ ﾁ03 71 ｨｦ ﾁ ﾁ ｨｪ0708ｨｦ 02 ﾁ ｨ ﾁ ﾁ ﾂ ﾁ0703 ﾁ ! ｨ ﾁ , ｨｬ , , ﾁ ﾁ ﾁ 08 ﾁ ｨｦ06 07 ﾁ ﾁ0302. ｨ ﾁ ｨｦ ﾁ ﾁ ﾁ ｨ ﾁ ﾁ ! ﾁ ﾁ ﾁ ｨ ﾁ ﾂ ﾁ ｨｦ. 05. ｨ ﾁ09 ﾁ ﾁ , , , ﾁ, : ﾂ., ﾁ09 ﾂ.00., ﾁ , ｨ.ｨｪ. 07 ｨｬ ｨｬ., ｨｪ ｨ., ﾁ09 ﾂ.03., ﾁ09 ｨ.05., ﾁ0308 ｨｪ.ｨ., ﾁ0308 ｨｪ. ﾂ ﾁ09 ｨ. ﾂ., ｨｮ ﾁ09 ﾂ.ｨ., ｨｮ ｨｰ.ｨ., ﾁ09 ｨｰ.ｨｮ ﾁ ｨｦ ﾁ ｨ ﾁ0308 ﾂ ｨｦ ﾁ ﾁ0202 罟ｪ02 ﾁ , ﾁ , ﾁ ｨｦ ﾁ ｨ ｨｦ ﾁ ﾁ0203 (07. 74{75). ｨｮ 綵 ﾁ "? ﾂ , ﾁ !1 7 ( {212), ﾁ ｨｦ ﾁ , , ﾁ , ﾁ ﾁ0205 ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ 縺 ｨｦ. 05. ｨ ﾁ ﾁ ﾁ ﾁ02 ﾁ030209, ﾁ ﾁ ﾁ ﾁ07 ﾁ ﾁ ﾁ ﾁ02 ﾁ ﾁ 罟ｪ ﾁ [13] ( ), ﾁ [15] ( ) ﾂ ﾁ , ﾁ09 ﾁ 縺 ﾁ !1 7, ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ , ﾁ ﾁ ｨｰ ( ), ﾁ ﾁ ﾁ ﾁ ﾂ , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ09 ｨｦ ﾁ ｨ ﾁ , ﾁ ﾁ 縺 ﾁ , 縺 ﾁ ﾁ ﾁ ﾁ 07 ﾁ [2, 07. III, ]. ﾂ ﾁ ｨ ﾁ0206, , , ﾁ , ｨｪ0708ｨｦ (07. 61), ﾁ ﾁ ｨｦ ﾁ (07. 63), 02 ﾁ ﾂ ｨｮ ﾁ ﾁ ｨ ﾁ ( ) ｨｮ ｨｦ. 05. ｨ ﾁ09 ﾁ ( ). ｨｮ ﾁ ﾁ ﾁ ( ), ﾁ 罟ｨ

74 ｨｦ ﾁ 罟ｪ ﾁ ﾁ ｨｦ ﾁ [3] ﾁ0902 ﾁ ( ) 罟ｪ ﾁ ﾁ ﾁ ｨｦ. 05. ｨ ﾁ03 ( ﾁ ﾁ ) , ﾁ ｨｦ (1954, 6 5 5), ｨｮ. ｨｦ ﾁ ﾁ (1964, 6 5 3), ﾂ. ｨｪ (1969, 6 5 3; 1974, 6 5 2), 02. ｨｮ. ｨ ﾁ09, ｨｪ. ｨ ﾁ09 07 ｨｰ. ｨ. ｨｮ (1984, 6 5 5), ﾂ (1987, 6 5 4; 2004, 6 5 7), ｨ. ｨｪ ﾂ ﾁ09 (2004, 6 5 5), ｨｪ. ｨｪ. ｨｰ ﾁ ｨ. 04. ｨｨ (2004, 6 5 5; 2005, 6 5 2), ｨ. ｨｦ. ﾂ ｨ ﾁ09 (2004, 6 5 5, 10) ﾁ ｨｮ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ｨｲ ﾁ03 ﾁ ﾁ03 ﾂ ( ﾂ 縺 ﾁ ﾁ ; 07. ｨ ﾁ09, ﾁ09, 01. ﾂ ﾁ, 00. ｨｦ ﾁ09 07 ﾂ. ﾂ ﾁ [8, c. 33{35, 180{183, 557{559, 629{631, 730{731, 735{736]) ( ﾁ09, 01. ｨ. ｨｲ ﾁ ﾁ09, 01. ﾂ ﾁ, 00. ｨｦ ﾁ09, ｨ. ｨ ﾁ0003 ﾁ, ｨ. ﾂ ﾁ09, ﾂ. ｨ [7, { 101, 106{107, 158{159, 232{233, 592{593, 608{609]) ﾁ ﾁ ﾁ03 ﾁ ﾁ 縺0603 ﾁ ﾁ ｨｪ ﾁ ( ; ; ) ﾁ ﾁ ﾁ ｨｰ ﾁ , ﾁ ｨ ﾁ ﾁ : ﾁ ﾂ , ﾁ ﾁ , ﾁ , ｨ. 04. ｨｨ ﾁ , : ﾁ , ﾁ ﾂ. ｨｪ ﾁ09, ﾁ , ﾂ ﾁ ﾁ ｨｪ 縺 : ﾂ ﾁ ｨｪ0401 [5; 11];

75 ｨｦ. 05. ｨ ﾁ ﾁ03 73 ﾂ [4]; , ﾁ ｨ ﾁ09, ﾁ ﾁ 縺 ｨ ﾁ091 7 [14], ﾁ ﾁ ﾁ ﾁ ﾂ. ｨｮ. ｨｮ ﾁ ﾁ09 09 ﾁ ﾁ ﾁ ﾁ03 ｨ ﾁ070709, ｨｲ ｨｰ03 ﾂ. ｨ. ｨｮ ﾁ09, ﾁ 07 ﾁ0305 ﾁ ﾁ ｨ ﾁ ﾁ ; ﾂ [6] ﾁ 縺 , ﾁ ﾁ ﾁ ｨｦ ﾁ ﾂ ｨ ﾁ ｨｮ09 ﾁ ﾁ , ﾁ03 ﾁ ﾁ ﾁ ｨｦ. 05. ｨ ﾁ ｨｮ09 ﾁ ﾁ ﾁ ( ) , ﾁ ﾂ , ﾁ ﾁ , ﾁ ﾁ ﾂ ﾁ ｨｦ , ﾁ ﾁ ｨ ﾁ ｨｨ , ﾁ03 ﾁ ﾁ ﾁ03 ﾁ *** ﾂ ﾁ ﾁ , ﾁ , ﾂ ﾂ ﾁ0707 ( ﾂ ﾁ0707) ( ), ﾁ ( ) ﾁ ﾁ ﾁ ﾁ , , , ﾁ ｨ ﾁ09, ﾁ , ﾁ0203 ﾁ09 ﾁ ﾁ , , ﾂ ﾂ ﾁ ﾁ ﾁ , ﾁ : 罟ｰ , , ﾁ ﾁ ﾁ ﾁ ﾂ ,

76 ｨｦ ﾁ ﾂ ﾂ ﾁ0707 ( { ) 縺 ﾁ ﾁ ﾁ ﾁ ﾁ , , ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ ﾁ ﾁ 罟 , , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ09 ﾁ ﾁ090303, ﾁ ﾁ ﾁ ﾁ, ﾁ ﾁ , ﾁ ﾁ ｨｰ ﾁ ﾂ ﾂ ﾁ ﾁ 縺 ﾁ ﾁ [18] ﾁ ﾁ ﾁ, 縺 , , ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ 07 ﾁ ﾂ ﾂ ﾁ ﾁ ｨｪ ｨｪ. ｨ ﾁ09, ﾁ ﾂ ﾁ ｨｪ0401 ( ), ﾁ (0502 ﾁ ), ｨｪ ｨｪ. ｨ ﾁ09 ( ) ﾁ ﾁ 縺0603 ﾁ ﾁ ｨｪ ﾁ ( ) , ﾁ09 ﾁ ﾁ , ﾁ ﾁ ﾁ , ﾁ ﾂ ﾂ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ﾁ

77 ｨｦ. 05. ｨ ﾁ ﾁ ﾁ , ﾁ ﾁ0305 ﾁ , , , , ﾁ ｨ ? , , ｨｦ ﾁ ｨ ﾁ ﾁ ﾁ ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ ｨ ﾁ ﾁ , ﾁ ﾁ ﾁ01, ﾁ , ｨｦ ﾁ ｨ ﾁ [16, ]: ｨ ﾁ ﾁ , ﾁ, ｨｰ ﾁ ﾁ ﾁ0308 ﾁ ﾁ ﾂ0205 ﾁ ﾁ ｨｦ ﾁ ﾂ ﾁ. ｨ ﾁ ﾁ050802, ﾁ! ｨｲ ﾁ0301 ｨｪ ﾁ0308. ｨｦ ﾁ ｨ ﾁ ﾁ ! , ﾁ ｨｦ. 05. ｨ ﾁ , ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ ｨｲ ﾁ ﾁ ( ) ﾁ ｨｪ ﾁ ﾁ ﾁ (ｨｪ040701) : ｨｪ ﾁ09, ｨｰ , , ﾁ : 縺 ﾁ , ﾁ0907 縺 ｨｰ ｨｮ ﾁ : (495) , ｨ ﾁ ｨｦ : I / ｨｦ.ｨｦ ﾁ ｨｪ.: ｨｪ0708ｨｦ , ｨ ﾁ ｨｦ ﾂ / ｨｦ. ｨｦ ﾁ ｨｪ.: ｨｪ I, 2004 (0403 ﾁ , ﾁ ) ; 07. II, ; 07. III, ｨ ﾁ ｨｦ 罟ｮ03 ﾁ ﾁ ﾁ , ﾁ0902 ﾁ ｨ // ｨｪ ﾁ , ﾁ ﾁ ﾁ : ｨ ﾁ ﾁ03 ﾁ, ﾂ // ﾁ ｨｪ.: ｨｪ040701,

78 ｨｦ ﾁ ﾁ , ﾁ 01. ﾂ., ﾁ09 00.ｨｦ. 07 ﾂ ﾁ ｨｪ0401 // , ｨｪ.: ｨｪ040701, 07. 3{4. 6. ﾂ ﾁ : // {6, ｨｪ.: ｨｪ040701, ( : , , ). 7. ﾂ ｨｪ ﾁ09, ｨｪ ｨｪ. ﾂ. ｨｬ ﾁ09, 24{ : ﾁ. ｨｪ.: ｨｦ ﾁ03 ｨｪ ﾁ ﾁ , ﾂ : ｨｪ ﾁ09, ｨｪ ｨｪ. ﾂ. ｨｬ ﾁ09, 28{ : ﾁ. ｨｪ.: ｨｪｨ 05ｨｮ , ﾁ ﾂ., ｨｮ ﾁ ﾂ.ｨ. ｨｪ ﾁ 08 ﾁ ﾁ. ｨｪ.: 縺 , ｨ.ｨｪ. ｨｰ ﾁ ｨｪ.: ﾁ , ﾁ09 00.ｨｦ., ﾁ , ｨｲ ﾁ ﾁ09 01.ｨ., ﾂ.ｨ., ﾁ 01. ﾂ. 07 ﾂ ﾁ ｨｪ0401 // , ｨｪ.: ｨｪ040701, 07. 3{ ﾁ09 00.ｨｦ. 罟ｦ ﾁ ﾁ ﾁ ﾁ : ｨ ﾁ ｨｦ ﾁ ﾁ // ｨｦ ﾁ0707 ｨ ﾁ: / ｨ. ﾂ ﾁ0308. ｨｪ.: ｨｦ0302- ﾁ03 ｨｪ040701, 2011, { ﾁ09 00.ｨｦ. ｨｪ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ : ｨｪ.: , (ｨｮ 縺 , , ) ( ); ｨｪ.: 罟ｬｨｦ09ｨｰ070507ｨｪ1 7, ﾁ09 00.ｨｦ., ｨｮ ﾁ ﾁ ﾂ.ｨｮ., ｨｮ ﾁ ﾂ.ｨ ｨ ﾁ09 // (1849) ｨｪ.: ｨｪ040701, ﾁ09 00.ｨｦ ﾁ ﾁ ﾁ ﾁ ﾁ : ｨｪ., 2006, ｨｮ ﾁ ﾂ.ｨ. ﾂ ﾁ , ｨｦ.05. ｨ ﾁ03, ﾁ , , ﾁ ｨｪ.: 0802ｨｰ ｨｮｨｹ, ｨｮ ﾁ ﾂ.ｨ. ｨｦ ﾁ09 ｨ ﾁ / ﾂ ｨｪ07 ｨｪ , 2006, 6 5 6, { ﾂ. ﾂ ﾁ ﾁ ｨｪ.: ｨｦ ﾁ03 罟ｦ , ﾂ. 06. ｨｦ. 05. ｨ ﾁ (1894{1975) // ｨｪ ﾁ ﾁ ｨｦ ﾁ ﾁ0209, ﾁ ｨｦ ｨｪ ｨｪ. ﾂ. ｨｬ ﾁ09, , [email protected]

79 13ｨｦ ﾁ ｨｲ " ｨｪ ﾁ ﾁ , ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｲ " ﾁ ﾁ ｨｪ ﾂ ﾁ ﾁ , ﾁ ﾁ ﾁ ｨｪ ", ﾁ ﾁ " ﾁ ﾁ0309: ｨｪ ﾁ , ﾁ ﾁ " : ｨｪ0006ｨｪ07: ｨｪ ﾁ09, ﾂ ﾁ , ｨｦ ﾁ03: ｨｪ ﾁ09, ｨ ﾁ , , , , ﾁ ( ﾁ ﾁ (499) ). ﾂ : ｨｪ.: ｨｦ ﾁ03 ｨｪｨ 08ｨｰ07", ISBN (091)

80 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ03 ﾁ ﾁ ﾁ ｨｰ , ﾁ03 ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ02, ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ0901 ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ , ﾁ ﾁ ﾁ 0803 ﾁ : (495) ｨ : ﾁ ｨｪ ﾁ , ﾁ0803 ﾁ ｨｦ : ｨｰ : , ｨｪ ﾁ " ﾁ ﾁ ﾁ PDF ｨｮ ﾁ ( ﾁ ) { ﾁ ﾁ ﾁ , ﾁ ｨｮ , ﾁ ﾂ , ﾁ ｨｪ ﾁ ", , ﾁ ﾁ. ｨｰ0309 ﾁ : : ｨｦ ﾁ ﾁ ｨｰ : 06/ ﾁ 07ｨ ｨｰ0903 ﾁ ｨｮ ", 05. ｨｪ ﾁ09, 09/ , 09ｨｦ , ﾂ ﾁ ﾁ ﾁ ﾂ ﾁ ﾁ ( ﾁ ). ﾂ ( ﾁ ) , ﾁ ﾁ ﾁ , ｨｮ ( ) ｨｰ ﾁ ﾁ , ﾁ ﾁ ﾁ Word, PDF ﾂ ! ﾁ ﾁ ﾁ (09 ﾁ ﾁ) ﾁ ｨｦ ﾁ ﾁ , ﾁ020403, ﾁ (06ｨｹ09) 07 ｨｰ ﾁ (ｨｰｨｦ0600). ｨｰ ﾁ0303 ﾁ ﾁ ﾁ ﾁ ﾁ. ｨ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ﾁ ｨｪ ﾁ ﾁ ﾁ ﾁ.

81 13Mathematical Education 6 5 1(73), 2015 Contents A. Ryazanovsky. Polynomials of One Variable 2 An introduction to polynomials theory for high school students. E. Skvortsova. Trigonometric Polynomials 12 An introduction to the theory of trigonometric polynomials for high school students, with a set of exercises. A. Evnin. Barycentric Method, Problems 27 A set of problems on barycentric method, with solutions. V. Ivlev. On a Representation of Sample Distributions 48 An analytic method of sample distributions representation is proposed, based on Fourier expansion w.r.t the Laguerre polynomials. E. Petrova, S. Pirogov. Dierent Addition of Dierent Quantities 53 It is shown that addition of dierent quantities in dierent mathematical or physical theories can be made in dierent ways. K. Lesan. What is Vector? 56 In the paper printed for the rst time in 1913 the idea of vector is discussed. This approach might have an inuence on the notion of vector which was introduced in Russian schools in 1970-s. T. Kuznetsova. On the 55-th Anniversary of the I. Andronov's seminar Progressive Ideas of Math Teaching in Russia and Abroad" 60 The history of the seminar grounded by the famous methodologist of math teaching Ivan Andronov. Information 77