ii95 



Inquiry into the Validity of a Method recently proposed by 

 George B. Jerrard, Esq., for Transforming and Resolving 

 Equations of Elevated Degrees : undertaken at the Re- 

 quest of the Association by Professor Sir W. R. Hamilton. 



[1.] It is well known that the result of the elimination of Xy 

 between the general equation of the m^^ degree, 



X = ^"^ + A a;™-' + B a;'»-2 + C x""-^ + D ^'"-^ 



+ Ea-'"-^ + &c. = 



and an equation of the form 



y = f{^), (2.) 



(in which / {x) denotes any rational function of x, or, more ge- 

 nerally, any function which admits of only one value for any 

 one value of x,) is a new or transformed equation of the m**" de- 

 gree, which may be thus denoted, 



{y -/(^i)} {y -/(^.)} -"{y -/W) = o, . . (3.) 



x^, x^, . . . x^ denoting the m roots of the proposed equation j 

 or, more concisely, thus, 



Y = y"* + A'^'"-^ + B'j/"'-2 + C'y'"-^ + D'y'"-'* 



+ E'^/"*-^ + &c. = 0, 



the coefficients A', B', C, &c., being connected with the values 

 f{x-^,f{x^, &c., by the relations, 



-A' =/(^i) +/(^,) + &c. +/U-J, 



+ B' =/(^i)/(^2) +/(^x)/(a^3) +/(^2)/(-^3) + &C. 

 -C'=/(^l)/(a;2)/(^3) +&C. 



And it has been found possible, in several known instances, to 

 assign such a form to the function / ix) or y, that the new or 

 transformed equation, Y = 0, shall be less complex or easier to 

 resolve, than the proposed or original equation X = 0. For ex- 

 ample, it has long been known that by assuming 



y=/(-^)=^^+^> (^0 



(5.) 



