40 Myr. Jerrard’s Remarks on My, Harley’s paper on Quinties. 
between functions of the form 
On, f(ar)(e4) 
and those of the class 
u in 
(t,t Vagt. BeOS) aa 
a complex process of a peculiar character must take the place of 
the simple method of substitutions. 
Thus from (v,) we see that 
©, esas) =n, BE) (18): 
and analogously from (w,), that 
©0090) Ond Be) (12) § 
in each of which equations the © of one member is accented 
differently from the © of the other in consequence of a transfer 
of ©! and ©” from branch to branch for the affix (Be) (78). 
This shows the necessity of carefully distinguishing between 
On, gab)(ea).. and (@,, 7) (ab) (ed) tS 
if we would avoid admitting relations among the roots through 
equations of the class 
(BM AE)(1) = On, Be) (12)- 
I might go on to elicit other properties of the theorem (vy, w), 
but what I have already said is, I think, sufficient for my purpose. 
6. Turning now to Mr. Harley’s paper, I find that the very 
existence of the theorem (v, w) is ignored. No wonder then 
was it—where so much circumspection was needed, and such a 
safeguard against fallacies as the theorem (v, w) was flung aside 
—that he as well as Mr. Cockle fell into error. 
7. But Mr. Cockle, not reflecting that the functions 
Pn(L1, Lay + +25) Xn(#yy Loy +» 5) 
(wherein z,, 2,..@, are known to be such as not to involve any 
irreducible radical of the form 4/2, while z,, 2,..2, are not 
proved to be subject to any condition) do not come within the 
scope of Lagrange’s theory for expressing one of two hLomoge- 
neous functions of the roots of a given equation in rational terms 
of the other, attempts by means of that theory to apply cbjcctions 
drawn from the failure of the method given m Mr. Harley’s 
aper to the method of solution in my ‘ Essay.’ The nature of 
Mr. Cockle’s second error is now manifest. He implicitly assumes 
