298 SIXTH REPORT — 1836. 



out being obliged to resolve any equation higher than the fourth 

 degree, and has even eflfected the transformation (12) without 

 employing biquadratic equations. His method may be described 

 as consisting in rendering the problem hideterniinate, by as- 

 suming an expression for y with a number of disposable coeffi- 

 cients greater than the number of conditions to be satisfied j and 

 in employing this indeterminateness to decompose certain of the 

 conditions into others, for the purpose of preventing that eleva- 

 tion of degree which would otherwise result from the elimina- 

 tions. This method is valid, in general, when the proposed equa- 

 tion is itself of a svjfficiently elevated degree ; but I have found 

 that when the exponent m of that degree is heloiv a certain 

 minor limit, which is different for different transformations, (be- 

 ing = 5 for the first, = 10 for the second, = 5 for the third, 

 and = 7 for the fourth of those already designated as the trans- 

 formations (12), (15), (18) and (20),) the processes proposed by 

 Mr. Jerrard conduct in general to an expression for the new 

 variable y which is a multiple of the proposed evanescent poly- 

 nome X of the m"^ degree in x ; and that on this account these 

 processes, although valid as general transform,ations of the 

 equation of the m^^ degree, become in general illusory when they 

 are applied to resolve equations of the fourth and fifth degrees, 

 by reducing them to the binomial form, or by reducing the 

 equation of the fifth degree to the known solvible form of De 

 Moivre. An analogous process, suggested by Mr. Jerrard, for 

 reducing the general equation of the sixth to that of the fifth 

 degree, and a more general method of the same kind for re- 

 solving equations of higher degrees, appear to me to be in ge- 

 neral, for a similar reason, illusory. Admiring the great inge- 

 nuity and talent exhibited in Mr. Jerrard's researches, I come 

 to this conclusion with regret, but believe that the following 

 discussion will be thought to establish it sufficiently. 



[2.] To begin with the transformation (12), or the taking away 

 of the second, third and fourth terms at once from the general 

 equation of the »*'*' degree, Mr. Jerrard effects this transforma- 

 tion by assuming generally an expression with seven terms, 



J/ = / {x) = A' x"-' + A" x"-" + A"' x"-'" 



+ M'a:^' -H W n^" + W» xf'" + W af"^^ .. (21.) 



the seven unequal exponents x' A" x"' ju.' /x" ju,'" /x'^ being chosen 

 at pleasure out of the indefinite line of integers 



0, 1, 2, 3, 4, &c. . . . . . . (22.) 



and the seven coefficients A' A" A'" M' M" M'" M'^, or rather 

 their six ratios 



