343 



the form £ , , considered as depending on t and on 

 g, ft, f ' 



a; . .... a; ; their initial values, and initial rates of increase 

 g»i' g,»' 



(relatively to /), namely £ and £' , are regarded as 



arbitrary but given and real functions of x JS . . .x j it is 



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

 sums of the forms 



S A, 



' V. g gAS V g g>*7 -4? \ g g> 1J 

 (A x ) n , 



\ g g,n) 



(3) 



are independent of g, and are == when any one of the ex- 

 ponents a., ...a 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 



s 3 A. = 2. * H, . A. ; (6) 



r h,r (1)1 h, i i, r * v * 



H..al (r + r'(A x J] vers fs *«.A x \ (7) 



h,h Ag\ K g g,h' J \ (i)l i g g,ij> v ' 



H,..=:2 A r'A x .A x . vers (s,. N w .w.A x .); (7V 

 h,% bg g g,h g g,t \ (01 * g g> l J v ' 



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* 2 , and of the 

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



the system of the n equations of the form 



