42 REPORT——1846. 
branches of science*, But as in this country they seem to have been thought, 
and by men not apparently unqualified to judge, to contain great additions to 
our knowledge, I cannot avoid inquiring whether this be true. 
Mr. Talbot points out in the early part of his first paper, that if there are 
nm — 1 symmetrical relations among the m variables 2, y...2, then the iden- 
tical equation 
{y---2}dx+(u...z)dy+t..,+ fy.-.dz=d{ry,..2} 
will assume the form 
g(x)dxu+ol(y)dy+...+¢(z)dz=d{xy...2}, 
and thus give us 
Sodet+foydy+..+fo(desay...2+C 
Precisely the same remark, though expressed in a different notation, is the 
foundation of M. Hill’s memoir, published in 1834, on what he calls “ func- 
tiones iterate.” It will be-found in Crelle’s Journal, xi. p. 193, A much 
more general theorem might be established by similar considerations: they 
are of course applicable whether the function ¢ be algebraical or trans- 
cendent. 
In the course of his researches, Mr. Talbot recognised the important prin- 
ciple, that the existence of 2 — 1 symmetrical algebraical relations among 
variables may be expressed by treating them as the roots of an equation, one 
of whose coefficients at least is variable, the others being either constant or 
functions of the variable one. Unfortunately he did not pass from hence to 
the more general view, that the existence of » —p symmetrical relations 
may be expressed in a similar manner if we consider p of the coefficients of 
the equation as arbitrary quantities. Had he done so, it is possible, though 
not likely, that he would have rediscovered Abel’s theorem; but as it is, he 
has never introduced, except once, and then as it were by accident, more 
than one arbitrary quantity. Thus only one of his variables is independent, 
and consequently, in more than one instance, his results are unnecessarily 
restricted cases of more general theorems, 
The character of his analysis will be perceived from what has been said. 
If / Xdz be the transcendent to be considered, X being an algebraical func- 
tion of «, he makes the following assumption— 
X=f(rv), 
v being a new variable, and fa rational function. From this assumption he 
deduces an algebraical equation in , the coefficients of which are rational 
functions of v. This equation then is one of those of which we have spoken, 
by means of which the function to be integrated can be expressed in a ra- 
tional form, Taking the sum with respect to the roots of this equation, we 
get 
2(Xd2)=3(f(«v) dz). 
It must be remarked that many forms might be assigned to the function f, © 
which would give rise to a difficulty, of the means of surmounting which 
Mr. Talbot has given no idea. If # and v are mixed up in f(a w), it is ma- 
nifest that we cannot integrate f(2v)d«, since v is a funetion of 2, which 
* It must be remembered also that Mr. Talbot admits himself to have been anticipated 
to a considerable extent by the publication of Abel’s theorem, 
