18 
tive ; and that when this quantity is negative, the expres- 
sions for the roots are 
x 0 =i(-b- T s *), 
ai=i(— 6— T*+T,*+ T s *), 
A— i( — M-T/+ T 2 '+ T 3 “), 
x 3 =i(-i+.T, i + T a * — T/). 
By the aid of the above results we can readily write out the 
explicit form of any one of the roots in terms of the coeffi- 
cients of the given qnartic equation. 
The success of the method is due to the circumstance 
that we can form, in the case of the quartic, a function of 
four letters which has only three values, and in the case of 
a cubic, a function of three letters which has only two 
values. But the method fails in the case of the quintic 
because we cannot form a function of five letters which has 
either only four, or only three values. If we employ 
Jacobis notation* adopted by Chief Justice Cockle, and 
write (a, as) for 
Xq-\-clX x -\- .... -\-a n i, 
a being a given root of the equation 
a p-i+ a p-*+ .... + 1 — 0 . 
in which is a prime factor of n, then in the case of the 
cubic (n= 3, p— 3) though (a, as) is six- valued, yet (a, as) 3 is 
only two-valued, and in the case of the quartic (n = 4i,p = 2), 
though (a, as) is twelve-valued, yet (a, as) 2 is only three-valued. 
But in the case of the quintic (n= 5, p=5), (a, as) is 120- 
valued, and (cc,as) 5 is 24- valued. And although when this 
latter function is developed, and reduced by means of the 
equation a 5 =l, to the form 
(w — u 0 -\- au x -j— aht 2 -j— ct — j— cthq , " 
it is found that u 0 has only six values, or is a root of a sex- 
* The explicit forms of u are given in an Addendum to my Paper on the 
Method of Symmetric Products. ( Society's Memoirs , Series II., Vol. XV., 
p. 217.) 
