ON THE RECENT PROGRESS OF ANALYSIS. 43 
if we eliminate we merely return to our function X. We must therefore 
express Xf (xv)d x in the form Vdv, V being a function and, as Abel has 
shown, an integrable function of ». Abel has given formule by means of 
which this reduction may be effected in all possible cases. But there is no- 
thing analogous to this in the writings of Mr. Talbot, and consequently he 
could not, setting aside the defect already noticed, obtain results as general 
as many previously known. In Mr, Talbot's investigations, f(a v) dz is such 
that Bf (xv) dx may be put in the form— 
Vi 2{P edz} + V2 {P,xde} + &e., . 
$,%, %.x, &c. (of which ¢',x, ¢',2, &c. are the derived functions) being 
rational functions of z. Then 2¢ 2 =a rational function of v by a well- 
known theorem. Let the form of this function be ascertained, and let us 
denote it by %v. Then differentiating, 
L@'rdr=xy'vdv, 
and hence 
ZXde=iUf(«ev)de=[V,x'0+ Vix.vt+..] dv, 
and the second side of this equation is of course rational and integrable. 
But the form of the function f(«v) is unnecessarily restricted in order that 
this kind of reduction may be possible, Nevertheless, Mr. Talbot's papers, 
from their fulness of illustration and the clear manner in which particular 
cases of the general theory are worked out by independent methods, will be 
found yery useful in facilitating our conceptions of the branch of analysis 
which forms as it were the link between the theory of equations and the in- 
‘ tegral calculus. 
4 In Mr, Talbot’s second memoir (Phil. Trans. 1837, part 2. p, 1) he has 
5 applied his method to certain geometrical theorems, Three of them relate 
__ to the ellipse, and are proved by the three following assumptions :— 
4 — e272) 4 1— 3 
i — oa =} =1+v2, or= std ve 
=, or = : 
1 c Aig V4 73) 4 1 el 
__ These assumptions are all cases of the following— 
{inser iactes : 
= ? 
1— 2x? a+a'z2 
_ where @, a’, ¢, e' are arbitrary quantities. The results of this assumption 
_ are completely worked out by Legendre (Théorie des Fonctions Elliptiques, 
iii, p. 192) in showing how the known formule of elliptic functions may be 
_ derived from Abel’s theorem. Mr. Talbot's first theorem is a case of the 
_ fundamental formula for the comparison of elliptic ares. This remark has 
_ reference to an inquiry which Mr. Talbot suggests as to the relation in which 
his theorems stand to the results obtained by Legendre and others. 
Tn conelusion, it may be well to observe that Mr. Talbot has remarked 
_ that, apparently, a solution discovered by Fagnani of a certain differential 
equation cannot be deduced from Abel’s theorem ; but as this solution may 
be easily derived from the ordinary formula for the addition of elliptic in- 
Ani, TI. 
11. I now come to the history of researches into the properties of par- 
_ ticular classes of algebraical transcendents. The earliest, and still perhaps 
_ the most important class of these researches relates to the transcendents 
