3:28 SIXTH REPORT — 1836. 



will be presently explained,) into other homogeneous functions, 

 according to the general types, 



c(r) = c('')3,o + cM^^^ + c(y\,, + C(\„ \- . 



(202.) 



^0 ^ - 1, 1 * " o,t 



each symbol of the class 



A«,B(''),CW, ...tW, (203.) 



P'9 P,9 P<9 p,g 



denoting a rational and integral and homogeneous function of 

 the 2 m new quantities, 



a\, f4, . . . a'^, (204.) 



and 



a\,a\,...a\, (205.) 



which function is homogeneous of the degree jo with respect to 

 the quantities (204), and of the degree q with respect to the 

 quantities (205). By this decomposition, we may substitute, 

 instead of the problem first proposed, the system of the three 

 following auxiliary problems. First, to satisfy, by ratios of the 

 m quantities (204), an auxiliary system of equations, containing 

 Aj equations of the first degree, namely, 



A% = 0,A% = 0, ...A('^>),,o = 0; (206.) 



^2 equations of the second degree, 



B'2,o = 0,B\o = 0,...B(H,o = Oj .... (207.) 

 A3 of the third degree, 



C'3,o = 0,C\o = 0,... 0(^3)3^0 = 0; (208.) 



and so on, as far as the following hf — I equations of the ith 

 degree, 



T',,o = 0,T% = 0,...T(^j^-i) = (209.) 



Second, to satisfy, by ratios of the m quantities (205.), a sy- 

 stem containing /i^ + h^ + ftg + . . . + h( — I equations, which 

 are of the first degree with respect to those ?h quantities, and 

 are of the forms 



