342 
the form 5, ips considered as depending on ¢ and on 
He prred, 43 ae: initial values, and initial rates of increase 
(relatively to ¢), namely 5 oh 0 and & g,n,0? Ate regarded as 
arbitrary but given and real functions of © pret, it is 
also supposed, in order to simplify the question, that all the 
sums of the forms 
a a a 
by 1 n , 1 
“Ag R 4, 1) eee (4, oe a} 3 2,,R (eer 1) eee (3) 
a, 
(45 *s, n) 
are independent of g, and are =0 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 
n 
= t= >. A, , €08 (¢, 4+ st—3, ut, i); (4) 
ee ee 
Say Ane = bs (5) 
fas 2B 2 a5 (6) 
r hr i ene As Sie 
H, = 3,,(n-+R’(A, 2, ,)) vers (s ra uA x ;), (7) 
g bi 
es , th 
ag =2, ze DF pg Mei ig isk (z,; uy 4%; yA: all) oF 
the index r being any integer from 1 to”, and being intro- 
duced in order to distinguish among themselves the nm dif- 
ferent (and in general real) systems of values of s*, and of the 
n—1 ratios of Ayes Ayes Ays which are obtained by resolving 
the system of the ” equations of the form 
