METHOD OF TRANSFORMING AND RKSOLVING EQUATIONS. 297 



algebra, four terms at once from the general equation of the m^ 

 degree, or reduce it to the form 



¥=3/"' + £'3/"*-^+ &c. = 0, .... (15.) 



by assuming an expression with four coefficients, 



because the four conditions, 



A' = 0, . . (8.) 



B' = 0, . . (10.) 



C' = 0, . . (14.) 

 and 



D' = 0, (17.) 



would be, with respect to these four coefficients, P, Q, R, S, of 

 the 1st, 2nd, 3rd, and 4th degrees, and therefore would in ge- 

 neral conduct by elimination to an equation of the 24th degree. 

 In like manner, if we attempted to take away the 2nd, 3rd, and 

 5th terms (instead of the 2nd, 3rd, and 4th) from the general 

 equation of the wP^ degree, or to reduce it to the form 



^m ^ Qlym-S ^ ^,ym-5 ^ ^^^ = Q, . . . (18.) 



so as to satisfy the three conditions (8), (10) and (17)> 



A' = 0, B' = 0, D' = 0, 

 by assuming 



3, = P + Q A- + R.r2 + x3, . . (13.) 



we should be conducted to a final equation of the 8th degree ; 

 and if we attempted to satisfy these three other conditions 



A' = 0, . . (8.) 

 C^ = 0, . . (14.) 

 and 



D' - a B'2 = 0, (19.) 



(in which a is any known or assumed number,) so as to trans- 

 form the general equation (1) to the following, 



Y = ^"^ + B'y"-2 + «B'2^'"-^ +E'?/'"-^ + &c. = 0, (20.) 



by the same assumption (13), we should be conducted by elimi- 

 nation to an equation of condition of the 12th degree. It might, 

 therefore, have been naturally supposed that each of these four 

 transformations, (12), (15), (18), (20), of the equation of the m^^ 

 degree, was in general impossible to be effected in the present 

 state of algebra. Yet Mr. Jerrard has succeeded in effecting 

 them all, by suitable assumptions of the function 3/or/(a:), with- 



