METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 305 



A",,o = 0, B\o = 0, (62.) 



A\i = 0, B"i,, = 0, B\^ = ; . . . (63.) 



and may in general be treated as follows. The two conditions (62), 

 combined with the m equations of definition (46), will in gene- 

 ral determine the m + 2 ratios of the m + 3 quantities j^o^Pu ••• 

 jOj^_p a', a", a'"; and then the three conditions (63), com- 

 bined with the m equations of definition (47), and with the m 

 other equations (56), will in general determine the 2m + 3 ra- 

 tios of the 2m + 4 quantities g^, q^, ... q^_^,pp^^_^, p'o^P'u 

 "•p'^_i, M', M", M'", M'Vj after which, the ratio of A'" to 

 M^^ is to be determined, as before, so as to satisfy the remain- 

 ning condition C = 0. But because the last-mentioned system, 

 of 2 m + 3 homogeneous equations, (63) (56) (47), between 

 2m + 4 quantities, involves, as a part of itself, the system (63) 

 of three homogeneous equations (rational and integral) between 

 m— 1 quantities, q^, q-^, ... q^^^, we see that it will in general 

 conduct to the result which we wished to exclude, namely, the 

 simultaneous vanishing of all those quantities, 



9o=0, ?, = 0, ... y^_2 = 0, . . . . (64.) 



unless their nwnber m — 1 be greater than 3, that is, unless 

 the degree m of the proposed equation (1) be at least equal to 

 the minor limit five. It results, then, from this discussion, 

 that the transformation by which Mr. Jerrard has succeeded in 

 taking away three terms at once from the general equation of 

 the ?n* degree, is not in general applicable when that degree is 

 lower than the 5th ; in such a manner that it is in general inade- 

 quate to reduce the biquadratic equation 



.r^ + A ^ + B .«2 + C a;3 + D = 0, . . . (65.) 

 to the binomial form 



2^ + D' = 0, ...... (66.) 



except by the useless assumption 



y = 1, {x^ + Aa^ + B x^ + C x^ + B), . . (67.) 

 which gives 



y = (68.) 



However, the foregoing discussion may be considered as con- 

 firming the adequacy of the method to redtice the general equa- 

 tion of the 5th degree, 



.r^ + A cr* + B x3 + C A-* + D X + E = 0, (69.) 



to the trinomial form 



VOL. A^^1836. X 



