ENSEMBLE OF SYSTEMS. 53 



dp x d de d r=n de du r 



du y du y dq x du y r \ du r dq x 



r ?" ( ^ e du r \ d de _ du y 



Therefore 



d(p, ...p n ) __d(u, .. . Q 

 d(u, . . . u^) d(q, . . . q n ) 



and 



^. ^) 



These determinants are all functions of the <?'s alone.* The 

 last is evidently the Hessian or determinant formed of the 

 second differential coefficients of the kinetic energy with re- 

 spect to <?j , . . . q n . We shall denote it by Aj. The reciprocal 

 determinant 



which is the Hessian of the kinetic energy regarded as func- 

 tion of the p's, we shall denote by A p . 

 If we set 



e & = I . . . / e A p dp,...dp n 



+00 +00 Mj 2 . . . n 2 



f. . . C 



e 20 d Ul . . . du n = (27r) , (140) 



and *, = * - fe (141) 



* It will be observed that the proof of (137) depends on the linear relation 



du r 



between the u's and q's, which makes constant with respect to the differ- 



dq x 



entiations here considered. Compare note on p. 12. 



