358 Sir William Rowan Hamilton on Equations of the Fifth Degree. 



we shall have, by (20) and (30), the following system of expressions for the 

 functions v : 



'345 



= a*-5y', + dy",; 



V4.M = A* - 5¥'3 + dy' 



'534 



= a*-5y'4 + dy"4; 



and 



(113) 



(114) 



(115) 



D being still = w* — w' — w^ + «, so that d^ is still = 5. We have also the 

 equation : 



^2345 "T ^3254 "T ^4523 "l ^^5432 

 "r ^2453 T" ^4235 T" ^5324 T" ^3542 

 "T ^2534 "I" ^5243 "l" ^3425 "T ^4352 



^2354 I ^^3245 "T" •'^5423 T ^4532 



"T ^2543 + X5234 + X4325 + X3452 



T" ^2435 "T ^4253 T ^3524 l" ^5342 » 



because the first member may be converted into the second by interchanging any 

 two of the four roots x^, x,, x^, x^, on which (and on ^,) the functions x depend, 

 and therefore the difference of these two members must be equal to zero ; since, 

 being at highest of the fifth dimension, it cannot otherwise be divisible by the 

 function 



^=(x^- a?3) (x., — X,) (x^ - X,) (^3 — x^) (x^ — X,) (x, — X,), (116) 



which is the product of the six differences of the four roots just mentioned, and 

 is itself of the sixth dimension. We may therefore combine with the expres- 

 sions (113) and (114) the relations : 



^345 ~r Y453 + ¥534 =^ Y354 -f- Y543 -f- Y435 ; K^^t ) 



and 



y"3+y"4 + y",iz0. (118) 



30. With these preparations for the study of the functions v, or of any com- 

 bination of those functions, let us consider in particular the first of the three 

 following factors of the expression (104) for 54 x/h^ : 



