Sir William Rowan Hamilton on Equations of the Fifth Degree. 347 



BC li: — D. (55) 



If then we make, for abridgment, 



f - (0 - 0') D = (0 - 6') («' - «,' - 0,^ + «), (56) 



9 being still the same imaginary cubic root of unity as before, so that 



r = -15; (57) 



we shall have, in (39), 



r,^ + es' + e'c' = 10 - 2^, 

 D* + 0B* + 0«c* = — 20 + 20f , 

 B^c" + 0c'd^ + e^D^B" = 30 + lOf ; 



and, consequently (because bcd =. — 5), 



0A;=-lOO(5-f)(25a' + 2^) + 5OO(ll+f)«e; , ^^^^ 



(58) 



el= — 2000 (2 + f ) a ; 



while &^k' and GH' are formed from Ok and 61, by changing the signs of f . It is 

 easy, therefore, to see, by the remarks already made, and by the definitions (42) 

 and (43), that the quantities H3, h^, H5, Hg, when expressed as rational functions 

 of a, €, 7], I, or of x', p, q, r, will not involve either of the imaginary roots of 

 unity, 6 and w, except so far as they may involve the combination f of those 

 roots, or the radical -s/ — 1 5 ; and that Hj will be formed from H3, and Hg from 

 H4, by changing the sign of this radical. We shall now proceed to study, in par- 

 ticular, the composition of the quantity h^ ; because, although this quantity, 

 when expressed by means of a/, p, g, r, is of the thirtieth dimension relatively to 

 y, (p, q, and r being considered as of the second, third, and fourth dimensions, 

 respectively), while H3 rises no higher than the fifteenth dimension; yet we shall 

 find it possible to decompose h^ into two factors, of which one is of the twelfth 

 dimension, and has a very simple meaning, being the product of the squares of 

 the differences of the four roots x", x"', x^^, x^ ; while the other factor of h^ is 

 an exact square, of a function of the ninth dimension. We shall even see it to be 

 possible to decompose this last function into three factors, which are each as low 

 as the third dimension, and are rational functions of the five roots of the original 

 equation of the fifth degree ; whereas it does not appear that H3, when regarded 



2 Y 2 



