TRANSACTIONS OF THE SECTIONS. O 



variables instead of only three, and that I might suppose them con- 

 nected by the general equation, 



x" -^ a x"~^ + b x"~- 4- . . . . =0, 

 whose coefficients a, b, &c. are all different functions of an arbitrary 

 quantity r. Then, when any particular value is given to r, the n 

 roots of the equation (or in other words the n variables) become de- 

 termined as to their numerical value. 



And if r changes its value to another value, the n variables seve- 

 rally undergo a corresponding change. Therefore they all vary 

 simultaneously, and the variation of any one of them is a determinate 

 function of the variation of any of the others. 



The n variables being connected in this manner, I found that the 

 ' values of certain integrals which depend upon them might frequently 

 be shown to have an algebraic sum. That is to say, the equation of 

 condition between the variables being given, new properties of various 

 integrals were found to be deducible therefrom. But the inverse 

 problem was found to be much more difficult, namely, " When the 

 form of the integral was given, to discover the equation which ought 

 to be assumed between the variables." 



This problem is the more important one because it is what occurs 

 in practical applications of the calculus. The solution of it, which 

 I have given in the Transactions of the Royal Society for 1836, ap- 

 pears to me to be as simple as the nature of the problem admits of, 

 and it conducts readily and rapidly to the form of equation which 

 ought to be assumed in any particular instance. And the form of 

 that equation being known, the properties of the integral frequently 

 flow from it with a facility which is surprising, considering the na- 

 ture and difficulties of the inquiry. 



While I was occupied in this investigation, that distinguished ma- 

 thematician Mr. Abel published a very remarkable theorem, which 



/*Vdx 



gives the algebraic sum of a series of integrals of the form / — ■, 



t/ V R 

 when P and R are polynomials in x of any degree. 



The methods of reasoning by which he arrived at this theorem 

 appear to have been quite different from those which I pursued, and 

 the form of his solution is altogether different from mine, although 

 in all those instances which I have tried the results ultimately con- 

 cur (as might be expected) . But it will be observed that this cele- 



brated theorem is limited to those forms of integral / — j=, where 



V the polynomial R has a quadratic radical. 



m My method, on the contrary, applies with equal facility to the Cu- 



K bic Radicals and to those of all higher degrees, as well as to a great 



B many other integral forms of a more complicated nature. 



WL I have therefore proposed to drop the name which Legendre has 



^L given, of Ultra- Elliptic Integrals, since it appears that no line of di- 



^^k stinction can be drawn between them and integi'als in general, which 



^^B possess similar properties to an extent so much greater than has been 



^^H hitherto imagined. 



I 



b2 



