Sir William Rowan Hamilton on Equations of the Fifth Degree. 357 



twenty-four different values of Lagrange's function t, which has itself 120 

 values : but expressing now these values of t by the notation 



taicde = ">^^a + w"^* + "'''^c + "'"j^rf + WX„ (105) 



which differs from the notation (22) only by having lower instead of upper in- 

 dices of x; and is designed to signify that we now employ (for the sake of a 

 greater directness and a more evident generality) the five arbitrary roots x„ &c., 

 of the original equation ( 1 ), between which roots no relation is supposed to sub- 

 sist, instead of the roots x', &c., of the equation (2), which equation was sup- 

 posed to have been so prepared that the sum of its roots should be zero. 



29. Resuming, then, the calculations on this plan, and making for abridg- 

 ment 



A = Xa + Xi -{- Xe -{- a;a-\- x„ (106) 



so that — a is still the coefficient of the fourth power of x in the equation of the 

 fifth degree ; making also 



Vfaicde = iCa* X^ + 2Xa^ x/ -j- 4Xa^ X, X, + GXa" Xi^ X,-\-\ Ix^ Xj X^ X^, (10?) 



and 



Xjcde = 5 (Vf abode + ^bcdea + ^cdeai + ^fdeatc + ^eabcd) j ( 1 08) 



we find (because w* = 1), for the fifth power of the combination (105) of the 

 five roots x, the expression : 



^aicde = A^ -f ( w" — 1 ) Xicde +(«»'— 1 ) Xceid ] / ^09) 



+ (w — 1) Xedcft -f (ur^ — 1) Xdiec ; J 



and, therefore, for the six functions v, with the same meanings of those functions 

 as before, the formula : 



"Vcde^^ ■^{i^Kcde ~\~ i tcied-\- t^ldeie-\- ''ledci) I OlO) 



= A*-f-(«)-f «."- 2)Yed,-l-(a.'-+ ".'-2)y,„; J 



in which, 



4 Ycde ^^ ^icde + ^c2ed "T" ^deic "T ^edc2- \^^^) 



If then we make 



y^, = ^\ + y'\, y,3,= <,-y",, 1 



v,,,= y'3+v"3, Y,,, = y',-Y"„ , (112) 



Y,34 = y'4 + y'\, Y354 = Y'4 — y'\ ; 



