CEETAIN APPLICATIONS TO THE THEORY OF DEFINITE INTEGRALS. 771 
We thus fin d 
r Ax)dx 
r /(^) ^ 
]{x — a) ^^{x)<p^{x) 
— VJ 
_{x-a)(p^{x)_ 
aX log 
aQ + a^x.. / <Pci{x) 
Cq + C^X .. V <PiW 
Here the function to be developed is 
1^ •• _j_ * / 'Ps W 
«0 + « 
f{x) (gQ + g.a-..) \/(Pj(.^)-(co + Ci^--) 
(a:— «) ^ («o + «!«••) V'<Pi(^) + (co + eia?..) V '^^{x)’’ 
the ascending developments having reference to — «, and to the different simple factors, 
x—\, X — A 2 •• of The coefficient of in the first development is evidently 
The coefficients of 
/(«) 
^Pl(«)P2(«) 
I 1 
X — Aj’ X — h 
log 
(«o+aja ..) v/ipia + (Co + Cja ..) 'v/(p2(fl) 
- in the latter developments are 0. Hence we have 
2 
f{x)dx 
/(«) 
a) <Pi{x)(p^{x)' 
-Cl 
^Pl(«) P2(«) 
A^) 
l^{x — a) V'(P,('3^)P2W 
log 
log 
(gp + gifl..) \/(Pi(<?)-(go + Ci«.-) \^%[a) 
(gp + fflig'..) \/(P^{x) — {cq + c^x ..) \^(p^{x) 
{aQ + a^x..) ^^<p^{x) + {cQ + c^x ..) (p^ix)' 
which is Abel’s theorem. There is, however, nothing gained by the peculiar form in 
which it supposes the integral to be expressed. The resolution of the polynomial under 
the radical into two factors <pi{x), <p.J[X) is only a substitute, and an inconvenient one, for 
the more general hypothesis of the theorem of art. 22, which permits the function under 
the radical sign to be any rational fraction. 
25. The theorem of art. 22 is, I believe, more general than any which have been 
investigated with relation to the same well-marked class of transcendents. Beoch, 
JuKGEXSEX, and Minding have given formulae directly applicable to the case in which -4^ 
is a polynomial*. Their results agree in substance with the above, under the particular 
restriction supposed, but they are far more complicated in expression. The introduction 
of the symbol 0, definite in meaning and indicating the performance of operations which 
are always intelHgible and always possible, greatly simplifies the expression of general 
theorems. 
26. The most general form of the problem contemplated by Abel in his theory of the 
comparison of transcendents may be thus expressed. Required a finite expression for 
y)dx, f{x, y) denoting a rational function of x and y^ the latter of which quantities 
is itself an iiTational function of x given by an algebraic equation of the form 
. . . . • ( 1 .) 
wherein ^ 0 , are rational and integral functions of x^ the simultaneous values of .r 
in the different integrals being determined by an equation of the form 
y—r. 
* Ceelle’s Journal, vol. xxiii. 
5 H 2 
