576 



NOTES. 



we may find a relation between the Hessians of f with respect to the 

 x's and that with respect to the y's. Using the notation for Jacobians, 



by 21), 



23) 



fi 



But in every derivative 



8V 



ay 



Consequently the Jacobian on the right of 23) is the same as that on 

 the left of 21). 

 Thus we find 



3(u u] fci---Pi 2 <Uv.;*M. 2 



24) E yf=W7^--^ 



1 fta^-ftM, 



or the determinant of the transformed form is equal to that of the original 

 form multiplied by the square of the determinant of transformation. 

 If an ordinary form f vanishes for a particular set of values 



where the c's are not all zero, we can show that the form is indefinite. 

 For if we substitute for the x's excepting x n , the values 



25) i 



we have 



= n ls==n ] 



2 



26) 



a sn (c s 



X 



The first term, containing f(c i9 . . . c n ) as a factor, vanishes by hypothesis. 

 The sums in the other terms contain only n 1 variables. If then there 



are any values of t/ n ... y n \>> for which 



does not vanish, 



since ic n is independent of these variables, we may by suitably choosing 

 the value of x n make the form have either sign, it is therefore indefinite. 



