MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
193 
Now if it be considered that this method extends to any number whatever of 
variables, and that the coefficients of the equation may be any functions of each other 
that we please to make them, it will appear at once how wide a field of inquiry here 
opens before us. It was the wish to reduce these extensive but rather complicated 
results to something like a clear and connected system which obliged me to defer 
the publication of them longer than I should otherwise have wished, by which means 
I lost the priority which at one time was in my power of announcing the existence of 
this new branch of analysis ; for the results hitherto mentioned, together with many 
others, which for the sake of brevity I omit, were obtained in the years 1825 and 
1826, and consequently two or three years previously to the publication of Abel’s 
theorem. And it will be observed that they comprise large classes of integrals which 
are not contained in his formula 
P dx 
\/ R 
Of this I have given an instance in the integral, 
J d x x 6 — 1 , 
and the more general one, 
J* i x x n — 1 . 
But an unlimited number of such forms may be found by the method I have 
pointed out of “ changing the conditions” at first established between the variables. 
We may conclude therefore that if x, y, z are the roots of an equation of n di- 
mensions, having at least one variable coefficient, and if we can find the function 
tp x — — in terms of x, we may thence deduce the algebraic sum of the n integrals, 
J <j p x . d x + J'fyy • dy + 
But the inverse problem still remains. Given the function <p x, or the integral 
/V. . d x, to find the equation 
x n — p x n ~ 1 + + r = 0, 
of which x,y,z must he roots , in order that S J <p x . dx may have an algebraic 
sum P 
This is evidently the most important part of the subject, for in applying the method 
to practice the form of the function <p x is given beforehand. That this research 
requires methods of its own will appear at once from a simple example. 
Let J' J 1 + x n . d x be the proposed integral, where n is any whole number. Let 
us first suppose 
J l -f x n = <p x = d-' 
2 c 
MDCCCXXXVI. 
