3882 Mr. J. Cockle on Transcendental and 
ing relations between it and the « and 8 functions of art. 70 of 
my “ Observations” (vol. xvii. p. 508), viz. :— 
6;0,+ 0,0; +0,0,= a, 
0,0, + 0,0,+ 6,9,=P;, 
0;0,+60)+ 6,0,=8.; 
003+ 040, + 0,94 = ea, 
0;0,+ 005+ 9,0,;= Bo; 
and in like manner from* 
0.0,+ 6410 a4+2+ 00430044 
we deduce 
0,49 + 010+ 0;0,= e4, 
0:0, + 005+ 6,0,=e,, 
G,0q+t 0304+ 99, =Bs, 
0;03,+0,0,+ 0,0,=R4, 
0; 0+ 0.0; + 9,03= a3. 
A contemplation of these systems of equations seems to lead 
to the conclusion that a certain equation of the fifteenth degree, 
which Mr. Jerrard supposes to be capable of decomposition into 
five cubics, is not irreducible, but composed of a quintic and a 
10-ic factor. 
I cannot think that Abel’s conclusion is at all shaken by the 
researches of Mr. Jerrard. Of his cardinal proposition, the 
en 
Py tiFnagt=9 
of art. 106 of his ‘ Essay,’ Mr. Jerrard gives no proof. Direc- 
tions to compare (ab) and (ac) are insufficient instructions for 
attaiing a result fraught with difficulties so serious. For rea- 
sons already assigned, I believe the proposition to be erroneous, 
and incapable of proof. And with it the whole argument of Mr. 
Jerrard falls. 
I need not the warning from my own oversight to restrain me 
from dwelling unduly upon what I believe to be an error of Mr. 
Jerrard’s. But, with every recognition of his great claims to 
* Mr. Harley has completely determined these « and 8 functions; and I 
regret that he has not as yet published the whole of his investigations on 
quintics. The proposition of Mr. Jerrard, which Mr. Jerrard supposes that 
Mr. Harley has ignored, is, when considered in reference to processes which 
do not involve transformation to a soluble form, rather an axiom than a 
theorem. It may be stated thus: If © be equal to ©’, it is not in general 
equal to 6”. 
I would here add the expression of a hope that Mr. Harley, to whom I 
have communicated the above results on the theory of transcendental roots, ~ 
may soon publish some developments of them to which he has been led. 
