332 Mr. J. Cockle’s Note on the Remarks of Mr. Jerrard. 
formed equation. And, since 
Wr+Wy (pey and 6,+6,, and 6,0, 
are similar functions, if the 15-ic in the V of Mr. Jerrard be 
reducible to cubic factors, the 15-ics in 6,+ 6, and 6,6,, that is 
to say in € and 6, are so reducible. But this is not the case. 
Under the most favourable circumstances in which we can form the 
cubic in &, the coefficients are unsymmetric. And the structure 
of the 15-ic in y, which is reducible to a quintic and a 10-ic 
equation, discloses no means of attaining a cubic with known 
coefiicients. The most favourable combinations, those of the 
forms y-8.e, y;84%), OY ¥;8.83*, are unsymmetric. 
Further: the coefficients of the cubics of Mr. Jerrard are (see 
arts. 69, 94, 109, and 110 of his ‘ Essay+’) expressible rationally 
in terms of 2, Z,..%5, and the doctrine of similar functions 
shows that they are either symmetric or incapable of evaluation 
save by a quintic. In the former case the five cubics are iden- 
tical ; in both cases the results are illusory. It is a significant 
fact that the soluble form of art. 96 of my ‘ Observations,’ for 
which the sextic in ¢ degenerates into a cubic, is not irreducible t. 
The ‘Essay’ of Mr. Jerrard is of surpassing interest, but 
these objections to the particular portion of it which relates to 
the finite solution of quintics seem to me to be fatal. A deep 
admirer of his researches, and indisposed to regard as established 
conclusions in which Mr. Jerrard does not concur, I may be per- 
mitted to express a hope that the promised sequel to the ‘ Essay ’ 
will not be long delayed. 
Lastly : how can each one of the system of five cubics men- 
tioned in art. 110 (p. 84) of Mr. Jerrard’s ‘ Essay’ be separated 
from the rest save by a quintic ? How can this quintic be solved 
unless it be an Abelian? And how can it be an Abelian if the 
given quintic be not an Abelian? What evidence is there that 
the four N’s vanish as alleged by Mr. Jerrard ? 
4 Pump Court, Temple, London, E.C., 
April 7, 1860. 
* Mr. Harley has completely determined all the 8 and & functions. I 
have selected these combinations from his values, kindly communicated 
to me. . 
+ Part I. 1858; Part IJ. 1859. Taylor and Francis. 
} See a solution, for the case Q=1, by Mr. Stephen Watson in the 
‘Educational Times,’ April 1860. Neither of the standard particular sol- 
vable forms 
oa 2 #4+3=0, a&—5a?+2=0 
to which I have been conducted is irreducible. 
