METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 331 



otherwise, unless the number of its equations be less than m — 1. 

 For if we put, for abridgement, 



(220.) 



and 



a",_aa',= &j,a"2-aa'2 = Z»2,.-«"w_i-«a'«_i = *„,_!> (221-) 



we shall have, as a general system of expressions for the m 

 quantities (205.), the following. 



a'\ = aa\ + b^, a"^ = aa'c^ + b^ 



+ *m-l> «"m= ««'».; 



m-l = ««'«,-! 



(222. 



(223.) 



a"^ = a a: 

 and therefore by (197), 

 ai = (1 + a) a\ + b^, . . . a^_i = (!+«) a'^_i 



+ *m-l5 «m = (l + «)«'m5 



so that the homogeneous functions A^"\ B*^'^), . . . T^'') may be, 

 in general, decomposed in this new way, 



AW = (1+«)A^W+A^W; 



BC^) = (1 + ar B<^1 + (]+«) B<^^ + B^W 



> . (224.) 



tw = (1 + ay rw + (1 + a)^-i rw + 



each symbol of the class 



A<«), B^^'^), . . . T<^^ , . . (225.) 



P>9 P'9 P'9 



denoting a rational and integral function of the 2 m — 1 quan- 

 tities a\, . . a'^, Aj, . . b^_i, which is homogeneous of the di- 

 mension p with respect to the m quantities 



«'„..<, (204) 



and of the dimension q with respect to the m — J quantities 



*„--A^-l, (226.) 



but is independent of the multiplier a. And the identical equa- 

 tions obtained by comparing the expressions (202) and (224), 

 resolve themselves into the following : 



