cally, so as to express x l ... x n in terms of y l ... y B , will also be of the 



form x t =s ; where Y is a function of yi ... y n , which may be de- 

 dyi 



fined by the equation 



in which the brackets indicate that the terms within them are to be 

 expressed as functions of y l ...y n . Moreover, if'_p be any other 

 quantity contained explicitly in X (besides the variables x l ... #), 

 the following relation will subsist ; namely, 



+ *I=0. 

 dp dp 



the differentiation in each case being performed only so far asj^ 

 appears explicitly in the function. 



It is then shown that if X contain explicitly, besides x l ... a? n , the 

 n constants a lt a 2 , ... a n , and the variable t, and if the In variables 

 x l ... x n , yi ... y n , be determined as functions of t by the system of 

 2 equations, 



U2\. U A. i /yy \ 



-r-y^ -r-= b i ....... ("0 



tUTj da { 



where b l ... b H are n other constants, the elimination of the 2 con- 

 stants from these equations and their differentials with respect to t, 

 leads to the system of differential equations (I.), if for Z be put the 



JV 



result of substituting in the values of the 2 constants in 

 dt 



terms of the variables. The equations expressing the 2 constants 

 in terms of the variables may be considered as the 2n integrals of 

 the system (I.). 



The author employs the symbol [ />, q] in a sense similar to that 

 in which Poisson and others have employed (p, g), namely, as an 



abbreviation for S/fl d J-_^L. ^L) and he shows that if p, q 

 ^dy t dx t dXi dy/ 



represent any two of the 2w constants ^ &c., 6, &c., then [p, q] is 



either =1 or =0, according asp, q are a conjugate pair ,-, b it or not. 



Next it is shown that if a,, a 2 , ...a n represent any functions of 2nn 



variables x l ... x n> y, ... y n , satisfying identically the n ^ n ~ > coh- 



2i 



ditions [a it a~\=Q, then if by means of the given equations ex- 



