i6 



MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



1.370 Functional Determinants. 



If yi, y2, . . . ., y n are n functions of xi, x 2) , x n : 



Jk = 



the determinant: 



dyz dyz 



dyi 



dx n 



dy 2 

 dx n 



dxz 



(xi, ^ 2 , ... .,x n ) 



is the Jacobian. 



1.371 If yi, y 2) ........ , y n are the partial derivatives of a functior 



F(XI, Xz, ..... , X n ) I 



dF ,. 



y* = (* = 



the symmetrical determinant: 



H 



i dx 



(dF dF dF\ 



(dxi' dx 2 ' ' ' ' dxj 



d(xi,x 2) 



is the Hessian. 



1.372 If yij y 2 ; , y n are given as implicit functions of xi, x^, 



x n by the ft, equations : 



Fi(yi, y z , , y n , x i} x 2) , x n ) = o 



, x n ) = 



then 



,y n ) 



d(xi, X 2 , . . ., x n ) 



, 



,F 2 , . . ., F n ) ^ d(F lt F 2 , . . . , F n ) 

 , Xz, . . ., x n ) ' d(y 1} y 2 , . . . , y n ) 



1.373 If the n functions yi, y 2 , , y n are not independent of each other 



the Jacobian, /, vanishes; and if / = o the n functions yi, y 2) . . . ., y n are not 

 independent of each other but are connected by a relation 



F(yi, yz, , y n ) = o 



