364 Sir'William Rowan Hamilton on Equations of the Fifth Degree. 



T532, = 900 + 620«.^ — 515«»^ — 935«» — 70«.^ J 



but, by (20), (30), and (54), 



2 (a. + «.") = — 1 + D, 2 («)^ + «.^) = — 1 — D, D^ = 5 ; (163) 



therefore, 



T2453 + T3M2 = 0, t,,3,+ t,3^ = 2250-1000d; (164) 



and 



v,„ = ^(1125-500d). (165) 



35. To compute the third of the functions v, we have, in the present ques- 

 tion, the relations : 



'^2534 ''3254' ''"5-243 '^4235' ''"3425 '^5324' ''"4352 '''4523 ' (. ^'^"j 



and, therefore, by (159) and (164), 



y,3, = — 375 + 250D. (167) 



For the fourth function v, we have, by processes entirely similar to the forego- 

 ing : 



T«a4=-(l--o^(2 + «'0^ T,,3, = _(l_«,^)*(2 + ..)^ 1 



T2354 + T4532 = - 2250 - IOOOd ; j ^ ^ 



'3245 



T3245-fT,«3=+750; ] ^ "^^ 



V3^=z— 375 — 250d. (170) 



For the fifth function v, we have the relations : 



'''2543 ''"2354 5 ''"5234 -~ ''"4325 5 ^3452 ^^ T4532 5 \^' '■J 



and, therefore, by (168), 



v^3=i(1125 + 500d). (172) 



Finally, for the sixth function v, we have 



''"2435 — - ''"5423> ''"4253 ''"3524' ''5342 -— ''"3245 5 ( W <J ' 



and, therefore, by (I69), 



