240 



could not by any determination of C give each of the roots 

 y—yi, y—y2' The differential equation must have a term 

 independent of y, and the solution is 



2/=yi+C (1-yO 

 which in virtue of y^-^yo—2, is 



=y,+(2-C) (1-y,) 



and gives y^ or y.^ as 0=0 or C=f2." 



Mr. W. H. L. Russell, of Shepperton, Chertsey, has 

 favoured me with the following elegant solution of the 

 quartic resolvent obtained by the aid of definite integrals. 

 ( 5 2 1 /17 5\ /14 2\ yll _lx X 



5^ = A L , 4 ' 4 ' ~i 3 . VT'i j VT'i AT'~4 j ,^, , 

 I ^ "T" 3 • 2 • 1 "^ (6 • 3) (5 • 2) 4 • 1) ^ / 



/ 9 6 3 y21 9\/18 6\/15 3\ 



+bJ i'i'i ^3^vT'iAr;AT'4>>^ 



I ^ ^ 4 • 3 • 2 ^ (7-4) (6-3) (5-2) ^ 



/ 13 10 7 /^25 13x/^22 lO-v >^19 7>^ s 



^Qx' T'T'i 3 . vT'TyVT'Tyl^T'i) e. . 



(1+5 43''^ "l8~^ ) (7^1) WW ' 



=A I . 10485763:^ t f' f' ^ ^' ^^'^ (1—^)^ (l^^F (1— m^)^ <^t^ dw dz 

 ( 8197r=^ ' \—vzivx^ 



4-Ba; f'f'f' ^^ '^^ ^"^ (1—^)^ (l-~;s)^" (l--w)^ <^y d% dw 

 "^ l~-t;2z^a;3 



4-Ca;- f'f'f' ^^ ^' ^^ (1—^)^ ^—^ {\—v^ dv dz dw 

 ^ \—vzivx' 



In this solution x is assumed less than unity. An inter- 

 esting paper by Mr. Russell, on the Theory of Definite 

 Integrals, will be found in the ^^ Philosophical Transactions'* 

 for 1854, pp. 157-178. 



Dr. Boole has also pointed out to me a method of solving 

 the quartic resolvent, which does not, however, essentially 

 differ from the above. And Mr. Russell remarks that a 



