30G SIXTH REPOUT — 1836. 



3,5 + D> + E'=0; (70.) 



and to effect the analogous transformation (12) for equations 

 of all higher degrees : an unexpected and remarkable result, 

 which is one of Mr. Jerrard's principal discoveries. 



[5 .] Analogous remarks apply to the process proposed by the 

 same mathematician for taking away the second, third and fifth 

 terms at once from the general equation (1), so as to reduce that 

 equation to the form (18). This process agrees with the fore- 

 going in the whole of its first part, that is, in the assumption of 

 the form (21) ioiif{x), and in the determination of the five ratios 

 (31) so as to satisfy the two conditions A' = 0, B' = 0, by sa- 

 tisfying the five others (26) (27) (28) (29) (30), into which those 

 two may be decomposed ; and the difference is only in the se- 

 cond part of the process, that is, in determining the remaining 

 ratio (32) so as to satisfy the condition D' = 0, instead of the 

 condition C = 0, by resolving a biquadratic instead of a cubic 

 equation. The discussion which has been given of the former 

 process of transformation adapts itself therefore, with scarcely 

 any change, to the latter process also, and shows that this pro- 

 cess can only be applied with success, in general, to equations 

 of the fifth and higher degrees. It is, however, a remarkable 

 result that it can be applied generally to such equations, and 

 especially that the general equation of the fifth degree may be 

 brought by it to the following trinomial form, 



2,5 + c>2 + E' = 0, (71.) 



as it was reduced, by the former process, to the form 

 y5 + D'j/ + E' = 0. . . (70.) 

 Mr. Jerrard, to whom the discovery of these transformations 

 is due, has remarked that by changing y to — we get two other 



trinomial forms to which the general eqtiation of the fifth de- 

 gree may be reduced ; so that, in any future researches respect- 

 ing the solution of such equations, it ivill be jicrmitted to set 

 out xoith any one of these four trinomial forms, 



^5 + A .r-* + E = 0, "] 

 St" + B a^ + Ya = 0, 

 ^5 + C ^2 ^ E = 0, 

 x^ + !> X + E = 0, 



in which the intermediate coefficient A or B or C or D may evi- 

 dently be made equal to unity, or to any other assumed number 

 different from zero. We may, for example, consider the diffi- 

 culty of resolving the general equation of the fifth degree as re- 



(72.) 



