766 PEOFESSOE BOOLE 0]S^ THE COMPAEISOX OP TEAXSCE^T)E^'TS, TTETH 
the medium of an equation of the third degree with reference to would lead to results 
possessing the same degree of generality. Thus the equation 
=v-\- wx + cjf, 
which connects x with v and w, and constitutes when freed from surds an equation of 
the third degree \vith reference to x, might have been employed. I am not aware that 
the above forms have been employed before. Legeadee, in deducing the properties of 
Elliptic Functions from Abel’s theorem, sets out from a different assumption, and as I 
think a less simple one, leading however to equivalent results. 
It is not my intention to enter here into the subject of the connexion of the different 
solutions which may thus be obtained. The theory^ of that connexion can however pre- 
sent no difficulties to those who are acquainted mth the laboui's of Jacobi, Eichelot 
and others, upon the differential equations on the integration of which the doctrine of 
the comparison of transcendents, as contemplated from another point of riew. depends. 
19. Before applying the general theorem of transformation to the investigation of 
general formulae for the comparison of transcendents, I vill say a few words upon Abel'.s 
theorem, as rvell as upon the class of theorems to which it belongs. 
Abel’s theorem is virtually an expression for 
2 C f{x)dx 
}{x — a) .v/(p(a7)’ 
f(x) and (p(x) being polynomials, the simultaneous values of x in the several mtegrals 
being connected by the equation 
or 
where x{x) is not restricted to being a polynomial, but is a rational function of x. in 
terms of the coefficients of which the value of the integral sum is to be determined 
Abel expresses <p{x) as the product of two polynomials <P-,{x), a form which is 
obviously given to it in order to meet the case in which a rational fraction occurs under 
the radical sign, since we have " 
Beoch and others, including, I believe, Abel himself, have considered very fully the more 
general case in which the polynomial (p(x) is raised to any fractional power whatever, 
and to this case may be reduced the still more general one in which a rational fraction 
takes the place of the polynomial. The reduction is however obwously far more com- 
plex than in the simpler case in which the index is I intend here to discuss this 
problem in a form sufficiently general to render all such reductions unnecessary. 
20. Peoblem. — Enquired a finite expression for the integral sum 
^j<p^-^«dx, ( 1 .) 
