METHOD OP TRANSFORMING AND RESOLVING EQUATIONS. 



to the binomial form 



/+E' = 0, (168.) 



except by the useless assumption 



y = L(a^ + A^'' + B^' + Ca?' + D<» + E), . . (138.) 



which gives 



y = 0. . . . (139.) 

 [11.] A principal feature of Mr. Jerrard's general method is 

 to avoid, as much as possible, the raising of degree in elimina- 

 tion ; and for that purpose to decompose the equations of con- 

 dition in every question into groups, which shall each contain, 

 if possible, not more than one equation of a higher degree than 

 the first ; although the occurrence of two equations of the second 

 degree in one group is not fatal to the success of the method, 

 because the final equation of such a group being only elevated 

 to the fourth degree, can be resolved by the known rules. It 

 might, therefore, have been more completely in the spirit of this 

 general method, because it would have more completely avoided 

 the elevation of degree by elimination, if, in order to take away 

 four terms at once from the general equation of the mth degree, 

 we had assumed an expression with thirty-three terms, of the 

 form 



y =f{x) = M x^' + A" x"-" + A'" x^'" 



+ W x/*' + 

 + N' x'' + . 

 + S'x^' + . 

 + O' x"' + . 

 + W x'°'' + 



. + M^v xf' 



(169.) 



. + OvnX" 



and had determined the six ratios of A', a", A'", M', . . . MP^) 

 and the twenty-five ratios of N', . . . TI^"!, so as to satisfy the 

 thirty-one conditions 



■^1,0,0,0,0,0 — ^> -"2,0,0,0,0,0 = "i (1/0.) 



•^0,1,0,0,0,0 = 0, a 1,1^0,0,0,0 — 0, Bq2oqq(j = 0, . . 



^ 3,0,0,0,0,0 + ^ 2,1,0,0,0,0 + ^ 1,2,0,0,0,0 + ^ 0,3,0,0,0,0 ^* 0, 



•^0,0,1,0,0,0 = 0, j 



■^1,0,1,0,0,0 + B 0,1,1,0,0,0 =0, I 



B 0,0,2,0,0,0 ^= 0, I 



^ 2,0,1,0,0,0 + C 1,1,1,0,0,0 + ^0,2,1,0,0,0 = 0, J 

 Y 2 



(171.) 



(172.) 

 (173.) 



