METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 307 



duced by Mr. Jerrard's researches to the difficulty of resolving 

 an equation of the form 



x^ + x + 'E-O; (73.) 



or of this other form, 



x^ -x + 'E = (74.) 



It is, however, important to remark that the coefficients of 

 these new or transformed equations will often be imaginary, even 

 when the coefficients of the original equation of the form (69) 

 are real. 



[6.] In order to accomplish the transformation (20), (to the 

 consideration of which we shall next proceed,) Mr. Jerrard as- 

 sumes, in general, an expression with twelve terms. 



1 '/ „^ 



p =f{x) = A' x^ + A".r + A'"x'' 



+ M'^ + M"^"+M'"X'+ Mi^o;''" j- (75.) 



+ N' / + N" x" + W x'" + Niv x''^ + N^ x'^ ; . 

 the twelve unequal exponents, 



X', X", A-, ,*', f.", f.'", ^.^^ /, v", /", v^ vV, . . (76.) 



being chosen at pleasure out of the indefinite line of integers 

 (22) ; and the twelve coefficients, 



A', A", A'", M', M", M'", M^^ N', N", N'", W, N^, (770 

 or rather their eleven ratios, which may be arranged and grouped 

 as follows, 



M' M" m;;^ ,^ . 



Miv> M»^' M^v' ^' ^^ 



N 



V. (81-) 



A'" 



^. (82.) 



being then determined so as to satisfy the system of the three 

 conditions 



A' = 0, . . (8.) 



C' = 0, . . (14.) 



D'-«B'2 = 0, (19.) 

 x2 



