METHOD OF TRANSFORMING AND RESOLVING EOUATIONS. 325 



A' = 0, B' = 0, C = 0, 

 may be decomposed ; the symbol A'^ ^ ^ ^^ ^ ;, or B'^ ^ ^ j-^ /j.^ p or 

 C'^ „ ^ i jt I denoting here a rational and integral function of the 

 thirty-three coefficients A', . . . n"^", which is homogeneous of 

 the degree/ with respect to A', A", A'", of the degree^ with 

 respect to M', . . . M^^, of the degree h with respect to N', . . . N^, 

 of the degree i with respect to B',... ff^, of the degree k with re- 

 spect to O', . . . Ovn, and of the degree I with respect to n', . . . 

 nvill : while the remaining ratio of MIV ^^ n^ni, shoiUd after- 

 wards be chosen so as to satisfy the remaining condition 



D' = 0. 

 But, upon putting, for abridgement. 



+ 



■Po^ 



m — I TO — 1 TO — 1 



TO — 1 ' TO — 1 



TO — 1 TO — 1 « 



.„} 



TO— 1 TO — 1 -f 



Af l(^') + A" L^^") + A'" l(*"') + M' L^f"'^ + ... 



+ N' L('') + . . . + N^L('^) + S' L(?') -I- . . . 



+ O' L('') + . . . + O^" L^-''") + n' L^-O + . . . 

 + 11^ 



tViii l(-^'") = L, 



(180.) 



(181.) 



(182.) 



(183.) 



