Sir William Rowan Hamilton on Equations of the Fifth Degree. 353 



and 



l'„ = - 110/5 - 4^' + 350rjf, 



l', = — llOjoV + ^Qpq" — 275qs + 350r^ 



l'j =z — 17/5' — 2-5p* + 55qr, 



L'3 = + 31/ - I75pr + llOy^ 



l/,= -90pq; 



l"„ = — 45/* + 12^^ - 75?-* ; 



l", = — 45pV + 68^5^ — 350^* — 75r" ; 



l"j = + 64/y - 107 5ps + 195yr ; 



l"3 = — 6p^ — 90pr + 150^^ ; 



l", = + igOpq — 1750*. 



} (87) 



I (88) 



But because, after the completion of all these transformations and reductions, it 

 is seen that the five quantities 



5l' 



•^"0. 



5L',-fL"„ 5L',-fL"„ 5l'3-^l"3, 5L',-fL"4, (89) 



which become the coefficients of y, x'\ y, ,r'^ af\ in the numerator 5l' — ^l" 

 of the fraction to be squared in the formula (85), are not proportional to the five 

 other quantities 



r, 2q, 3p, 0, 5, (90) 



which are the coefficients of the same five powers of a/ in the denominator of the 

 same fraction, it may be considered as already evident, at this stage of the inves- 

 tigation, that the root .7/ enters, not only apparently, but also really, into the 

 composition of the quantity h^. 



26. The foregoing calculations have been laborious, but they have been made 

 and verified with care, and it is believed that the results may be relied on. Yet 

 an additional light will be thrown upon the question, by carrying somewhat far- 

 ther the analysis of the quantity or function H4, and especially of the factor l ; 

 which, though itself of the ninth dimension relatively to the roots of the equation 

 of the fifth degree, is yet, according to a remark made in the nineteenth article, 

 susceptible of being decomposed into three less complicated factors ; each of these 

 last being rational with respect to the same five roots, and being only of the third 

 dimension. In fact, we have, by (62), and by (11), (12), (13), 



2z 



vol. XIX. 



