Sir William Rowan Hamilton on Equations of the Fifth Degree. 355 



+ \Ax, {x^x^ + x^x,) — Qx, (x^ + 0:3) (x^ + X,) 



-{j;^'-^x,'+x,'-\-x,'-2(x^^+x,^)(x,+x,)-2(x^^+x,') (x^+x,)] ; . 



while the second factor, m^, can be formed from Mj by merely interchanging ^3 

 and x^ ; and the third factor M3 from m^, by interchanging x^ and Xy 



27. If, now, we substitute the expression (94) for the numerator of the frac- 

 tion which is to be squared in the formula (81), and transform also in like man- 

 ner the denominator of the same fraction, by introducing the five original roots 

 Xj, . . . x^, through the equations (77) and (4), we find : 



H4 = 



(•*"l -^2) (-^l ^3) {^1 •^4) {^1 ^5) 

 and we see that this quantity cannot be a symmetric function of those five roots, 

 unless the product of the three factors Mj, m^, M3 be divisible by the product of 

 the four differences a:, — x^ . . . x^ — Xy But this would require that at least 

 some one of those three factors m should be divisible by one of these four dif- 

 ferences, for example by or, — x^; which is not found to be true. Indeed, if 

 any one of these factors, for example, Mj, were supposed to be divisible by any 

 one difference, such as x^ — x.^, it is easy to see, from its form, that it ought to 

 be divisible also by each of the three other differences; because, in m,, we may in- 

 terchange x^ and Xj, or x^ and x^ or may interchange x^ and x^, or x^ and x^, if 

 we also interchange x^ and x^, or x^ and x^ : but a rational and integral function 

 of the third dimension cannot have four different linear divisors, without being 

 identically equal to zero, which does not happen here. The same sort of reason- 

 ing may be applied to the expressions (95), combined with (93), for the three 

 factors M„ M2, M3, considered as functions, of the third dimension, of a, j8, 7, 8 ; 

 because if any one of these functions could be divisible by any one of the four 

 following linear divisors, 



or, — a:^ = — 5a— (/3 + 7 + 8), 



x^ — x^= — 5a—(^ — y — h), 



Xi — x^= — 5a—(—p-\-y—d), 



Xi — x^^ — 5a 



(_p_7 + 8), J 



(102) 



2z2 



