202 Mr. J. Cockle on the Theory of Equations 
roots of a 10-ic equation, each root of which is a rational and 
symmetric function of two roots of the given quintic. 
95. Again, we may (art. 56) express 3, as a rational function 
of @,; and if for a moment we write 
d| _10/ E ph Dib aa See a 
ot =o t= V3; (cos pe Y¥—l sin *), 
c 1@ " ] {>be i] 
sf =o" t= /P=3,( cos st v—1 sin =): 
then 3, expressed in terms of 5, is 
2m 2m! . we ov 
9,=P+9,(1 + 2cos —")_2(P—s,) cos —— +46, sin Cee 
and if we elevate each side of this equation to the fifth power, 
expand and eliminate m and m! by means of 
! 
coom=—4,  cosm'= —-—, 
3° (P—3,)* 
we shall have one of the actual expressions on which the fore- 
going and (virtually) Mr. Jerrard’s argument is founded. As to 
my own particular view (arts. 58, 59), I may add that if 3, were 
a rational function of $,, the roots of the quintic would contain 
no quintic surds unless (which there is no reason to suppese, 
though I once suspected it) the theory of Abelian sextics is im- 
perfect. The error of Mr. Jerrard inheres, in my opinion, in his 
mode of comparing the equations (ab) and (ac) at pages 80 and 
81 of his most valuable ‘Essay.’ His functions 2, .&, 4, and 
= in art. 104 are foreign to the question, mere instruments for 
eliminating radicalities. They lead to no other result than that 
to which the immediate comparison of (ac) and 
B—)==0 
would conduct us, viz. an expression for & into which Pye.) 
enters irrationally. i 
96. The theory sketched in these papers has been developed 
in pages* more appropriate than the present to the details of 
mathematical processes. I would suggest that « may be ex- 
pressed as a rational function of y, and yw as an irrational func- 
* See a paper “On the Theory of Quinties,” by the Rev. Robert Har- 
ley, F.R.A.S. &c., in the Quarterly Journal of Pure and Applied Mathe- 
matics, January 1860. M. Wantzel’s argument will be found in M. Serret’s 
Cours d’ Algébre Supérieure (2me éd., Paris, 1854). 
