318 SIXTH REPORT — 1836. 



which should satisfy the two conditions 

 A' = 0, . . . (8.) 

 C' = 0, . . . (14.) 



and by then determining the ratio of A'" to N^, so as to satisfy 

 this other condition, 



E' = 0, ...._.. . (147.) 

 which could be done without resolving any auxiliary equation of 

 a higher degree than the fifth ; and this reduction, of the diffi- 

 culty of the sixth to that of the fifth degree, would have been a 

 very important result, of which it was interesting to examine the 

 validity. The foregoing discussion, however, appears to me to 

 prove that this transformation also is illusory ; for it shows that, 

 because the degree of the proposed equation is less than the mi- 

 nor limit 7) the functions ^ {x) and x (•*') i'l (^9) are connected 

 by a relation of the form (36) ; on which account the expression 

 (89) becomes 



y = f{x) = (A'" + a N^) cf> {x) + A N^ X, . . (90.) 



and the condition 

 gives, in general, 

 that is, 



E' = 0, . . (147.) 



(A"'+ aNV)'^ = 0, ..... (148.) 



A"' + aNV = 0; . . (92.) 

 so that finally the expression for y becomes 



y = \l^X, . . . . (93.) 

 that is, it takes in general the evidently useless form, 

 ^ = L (a;6 + A x^ + B ^-4 + C .r^ + D .r^ + E .r + F). (142.) 



[10.1 Mr. Jerrard has not actually stated, in his published Re- 

 searches, the process by which he would effect in general the 

 transformation (15), so as to take away four terms at once from 

 the equation of the m^^ degree, without resolving any auxiliary 

 equation of a higher degree than the fourth ; but he has suffi- 

 ciently indicated this process, which appears to be such as the 

 followino-. He would probably assume an expression with 

 twenty-one terms for the new variable, 



y=f{x)=A' A-"-' + A" x"-" + A'" X "■'" 



+ M' x''' + M" x^" + W" xf"'" + M'^ xf"'^ 

 + N' x"' + N" x'" + N'" x'"' + Niv ^''^ 



+ S' x^' + S'' x^" + S'" x^" + S"' *^" + S^ x^^ 



(149.) 



