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



Let us now compare these last numerical results with the general formulae found 

 by other methods in earlier articles of this paper. 



37. The method of the thirteenth article gives, in the present example, 



arr_-|-, )3 = 1, 7 = }, 8 = 0, e = ^^, ,j = 0, 



—[T^' ^ — 12 ' I — KA — y^, 



(184) 



^3+X3_^^^ l(,.3_X3)__2-5 3-.(e_^,). 



and, therefore, by (59), 



|g = 5(l_a S=12(2 + a I ^185) 



k' — SkP^X — P (k^ + \3) = _ 2^ 3' 5"" (23 + 3^) ; J 



and, accordingly, if we multiply the last expression (184) by the last expression 

 (185), we are led, by the general formula (60), to the same result for Vu^, and 

 therefore for H4, as was obtained in the last article by an entirely different me- 

 thod. The general formula (60) may also, in virtue of the equations (13), (59), 

 (62), (63), (70), (116), and (4), be written thus : 



18v/h, = — 5'»(0 — 0^)TirL; (186) 



which agrees, by (94), with the general result (145), and in which we have now 



Ti7 = 1 .2.3.1.2.1 = 12; (187) 



while L may be calculated by the definitions (62) and (63), which give, at pre- 

 sent, by the values (184) for a, e, 1/, i, 



M = f(l-rX "==-2(2+^), (188) 



and 



L=-^(23 + 3^): (189) 



and thus we arrive again at the same value of 's/h4 as before. The same value 

 of L may be obtained in other ways, by other formulae of this paper ; for example, 

 by those of the 24th and 25th articles, which give, in the present question, 



l' = — 2^ 3' 5^ 23 ; l"=z + 2' 3^ 5\ ( 19Q) 



We may also decompose l into three factors m, which are here : 



