178 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
The excellent work of Legendre* was destined to arrange, classify, and distin- 
guish the properties of elliptic integrals which are implicitly contained in Euler’s 
theorem above mentioned. In this treatise he has thoroughly examined the nature 
of these transcendents, and presented the results of his inquiries in a luminous and 
well-arranged theory. 
The extensive tables which accompany his work will enable future mathematicians 
to make as frequent and convenient use of elliptic integrals as they have hitherto 
done of circular and logarithmic ones. 
In the year 1828 Mr. Abel, of Christiania in Norway, published a very remarkable 
theorem, which gives the sum of a series 
of integrals of the form J' 
where P 
and R are entire functions of x, of the form x n + a x n ~ 1 + b x n ~ 2 * + ; n being 
any whole positive number, and a, b, & c. constant coefficients. 
This theorem extends much further than Euler’s, in as much as the latter is limited 
to those forms of R which contain no higher powers of x than the fourth. It departs 
still more widely from Euler’s theorem, in exhibiting the sum, not of two only, but 
of many integrals of the same form. And it must be observed that this plurality of 
terms 
is in general necessary ; for if we give to the expression J* 
d x 
—7~ its utmost 
generality, it does not appear possible to find the sum of only two such integrals in 
finite algebraic, or logarithmic terms ; but it is requisite to combine a greater num- 
ber of them, below which number the problem cannot be reduced. 
Abel’s theorem in general furnishes a multitude of solutions for each particular 
case of the problem : notwithstanding which it is possible to find other solutions 
which appear not to be comprised in his theorem, nor deducible from it-f. 
On the publication of this theorem the illustrious Legendre, who at an advanced 
age still cultivated his favourite science with all the ardour of youth, was one of the 
first to feel its extent and importance. And accordingly, with a degree of zeal almost 
unequalled in the annals of science, he devoted a large portion of time to the verifi- 
cation and elucidation of the theorem by numerical examples. The result of these 
calculations was amply confirmatory of its truth, and it therefore undoubtedly stands 
upon the basis of rigorous demonstration. 
There can be little doubt that the ingenious mathematician to whom we are in- 
debted for this theorem would have arrived at fresh discoveries, of not inferior value, 
' Exercices dc Calcul Integral. Paris, 1811. Traitd des Fonctions Elliptiques. Paris, 1825. 
• (l X (l 7/ • • • 
t For instance, if — + - — = 0, his theorem gives the integral .ry=:l ; but, apparently, it does 
Vl+vr' Vl +y 4 
not give this other integral y 2 
tiche, vol. ii. p. 3G9.). 
1 I | ^ o t 
= T. — , which was discovered by Fagnani (Produzioni Matema- 
v 1 + x* — x v 2 
