METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 319 



(153.) 

 (154.) 

 (155.) 



(156.) 



and would develop or decompose the coefficients A', B', C, of 

 the transformed equation in y, considered as rational and inte- 

 gral and homogeneous functions of the twenty-one coefficients, 



A', A'', A'", (150.) 



M', M", M'", W-^, (151.) 



N', N", N'", N^^ W, W\ (152.) 



S', B", S'", S^v, S^ S^i, «vn^ gviii^ , . 



into the following parts : 



■^ — -^1,0,0,0 + -^0,1,0,0 + -^0,0,1,0 + -^0,0,0,1 5 



" ~ ■" 2,0,0,0 + " 1,1,0,0 + -° 1,0,1,0 + -B 1,0,0,1 



4- Tl' 4- R' -4- Tl' 



^ -" 0,2,0,0 ^ -"0,1,1,0 ^ -° 0,1,0,1 



+ ■" 0,0,2,0 "*" ■" 0,0,1,1 + ■" 0,0,0,2 J 



^ — ^ 3,0,0,0 ^ ^ 2,1,0,0 "r ^ 2,0,1,0 "i" ^ 2,0,0,1 

 ' ^ 1,2,0,0 + ^ 1,1,1,0 + ^ 1,1,0,1 

 "I" ^ 1,0,2,0 "'" ^ 1,0,1,1 "I" ^ 1,0,0,2 

 ■t" ^ 0,3,0,0 + ^ 0,2,1,0 + ^ 0,2,0,1 

 + *-" 0,1,2,0 "f" '^0,1,1,1 "^ ^0,1,0,2 

 "'■ ^ 0,0,3,0 "t" ^ 0,0,2,1 + ^0,0,1,2 + C 0,0,0,3 5 . 



each part A';, -it, ; or B',^.^^^; or C';, • ;, ^ being itself a rational 



and integral function of the twenty-one quantities (150) (151) 

 (152) (153), and being also homogeneous of the degree h with 

 respect to the three quantities (150), of the degree i with re- 

 spect to the four quantities (151), of the degree k with respect 

 to the six quantities (152), and of the degree I with respect to 

 the eight quantities (153). He would then determine the two 

 ratios of the two first to the last of the three quantities (150) 

 (that is, the ratios of A' and A" to A'") so as to satisfy the two 

 conditions 



A'i,o,o,o = 0, B'2,0,0,0 = ; . . . . (157.) 

 the three ratios of the first three to the last of the four quanti- 

 ties (151), so as to satisfy the three conditions 



A'0,1,0,0 = 0, B'1,,^0,0 = 0, B'0,2,0,0 = ;' . . . • (158-) 



the ratio of the last of the quantities (150) to the last of the 

 quantities (151), so as to satisfy the condition 



^ 3,0,0,0 + C/ gj^QO + C 1^2,0,0 + C 0,3,0,0 ~ 5 • • • (159.) 



