34^ 



the form ? , . > considered as depending on t and on 



x ,y.. .X ; their initial values, and initial rates of increase 



(relatively to t), namely ? , q and ?' , are regarded as 



arbitrary but given and real functions of a? ,, ....r ; it is 



also supposed, in order to simplify the question, that all the 

 sums of the forms 



a 



^g K g g>^y \ g gy^J '% ^ g g^'' 



[A X )\ 



(3) 



are independent of ^, and are zz when any one of the ex- 

 ponents a,, . . . o is an odd number. These equations are 



analogous to, and include, those which M. Cauchy has con- 

 sidered in his memoir on the Dispersion of Light, and may 

 be integrated by a similar analysis. 



A particular integral system may in the first place be 

 found by assuming 



n 

 /a. = S,.,^ H, . A. ; (6) 



r h,r {1)1 ?i,t i,r ^ ^ ^ 



»M=2,/K + K'(A^^^,/)vers (s^,.;>^ A^.-^_^), (7) 

 H. .zzS. r'A X ,A X . vers (s,"w.A x . ) ; (TV 



fi,i ^g g g,h g gyt \ (01 2 g g,*J * ^ ^ 



the index r being any integer from 1 to n, and being intro- 

 duced in order to distinguish among themselves the n dif- 

 ferent (and in general real) systems of values of 5^ and of the 

 « — 1 ratios of a^, . • a^, . . a^^, which are obtained by resolving 



the system of the n equations of the form 



