MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
179 
if a premature death had not terminated his career, to the irreparable loss of science, 
at the early age of twenty-seven. 
Before concluding this slight historical sketch of the subject, I ought not to omit 
mention of a valuable recent memoir by M. Poisson*, in which he has considered 
various forms of integrals which are not comprehended in Abel’s formula. 
It has already been stated that the integrals to which Abel’s theorem relates are 
those comprised in the general expression where P and R are entirely poly- 
J VR 
nomials in x. Next in order of succession to these there naturally presents itself the 
class of integrals whose general expression is 
y'R 
where the polynomial R is 
affected with a cubic instead of a quadratic radical. 
But Abel’s theorem has no reference to these, and consequently affords us no 
assistance in their solution. The same may be said with regard to the succeeding 
classes of integrals, 77Tf’ an< J generally J 
V 
V If 
Still less does it enable 
us to find the sum of such integrals as J* <p (R) d x, R being as before an entire poly- 
nomial^, and <p any function whatever. This is the problem to the solution of which 
the following pages will be dedicated. 
I may be here permitted to mention, that Abel’s theorem was unknown to me 
until some years after its publication, and that these Researches were nearly com- 
pleted before I was acquainted with it. I have, however, made no alteration in them, 
but have chosen to present the subject in the manner in which it originally occurred 
to me. 
I am not aware that Mr. Abel has left any memorial of the successive steps of 
reasoning by which he arrived at his theorem. Probably they were very different 
from those which I have employed, and therefore I have detailed at some length my 
method of investigation, beginning with the first rudiments of the theory at which I 
afterwards arrived. 
§ 2 . 
It was remarked by the earliest inventors of the integral calculus, that there was 
a mutual dependence between the two integrals J "* y dx and J* x dy, so that if the 
one were given the other became known, by virtue of the equation 
/• dy +fydx = xy + C. 
If therefore one of these forms happened to be more easy of integration than the 
other, they directed it to be substituted for it. 
* Crelle’s Journal, vol. xii. p. 89. Berlin, 1834. 
t By “ polynomial” I here understand an expression containing at least two different powers of x. 
2 A 2 
