BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 355 



definite,* may be transformed! to real sums of squares of new variables A which are 

 real linear functions of />!... i/ n . 



Thus 2T = A 1 +A,+...+A. (17), 



= X 1 2 A 1 2 +X 2 2 A 2 2 +... + X B 2 A B 2 (18), 



X^.-.X,, 2 being the values of X which cause the vanishing of the determinant whose 

 (k, /)th element is 



_^_(V-X 2 T) (19). 



But by (3) and (16) this element is 



And the vanishing of the determinant is, therefore, the condition that the n body 

 equations 



ay OAi> * ... fa) = \* fa, h, ... fa) ...... (20) 



should have an integral other than zero. The integrals thus denned as to their body 

 values by (20) have already been denoted by P 1; P 2 , ... P n . 



The determinant being of the nth degree in X 2 vanishes for n values of X 2 . Since 

 one of the forms, T, is definite these n roots are all real (KRONECKER, loc. cit.), and 

 since both T and V are positive X/...X,, 2 are all positive ........ (21). 



Now let the coefficients in the transformation of T and V be the (p)'s defined by 



A! = hfapu+hfaPw+ infapin 



A B = hfap n i+^apa+---^fap m J 



Then differentiating (17) and (18) by fa and using (22) 



I "\rri 



(23), 



(24), 



for I = 1, 2 ...n. Now if we limit the hitherto arbitrary (t/)'s by making all the 

 (A)'s vanish except Ay, then 



..... (25), (26), 



* "Definite" here means: one-signed and vanishing only when f,, ^ 2 , ... $ all vanish. 

 t KRONECKER, quoted by BROMWICH, 'Cambridge Tracts in Mathematics,' No. 3, 26; also by 

 WEBSTER, ' Dynamics,' Appendix V. 



2 z 2 



