ON THE REGENT PROGRESS OF ANALYSIS. 251 



liis variables is independent, and consequently, in more than one 

 instance, his results are unnecessarily restricted cases of more 

 general theorems. 



The character of his analysis will be perceived from what 



has been said. If \Xdx be the transcendent to be considered, 



X being an algebraical function of x, he makes the following 

 assumption 



X=f(xv), 



v being a new variable, ajnd /a rational function. From this 

 assumption he deduces an algebraical equation in x, the co- 

 efficients of which are rational functions of v. This equation 

 then is one of those of which we have spoken, by means of which 

 the function to be integrated can be expressed in a rational 

 form. Taking the sum with respect to the roots of this equation, 

 we get 



It must be remarked that many forms might be assigned to the 

 function f, which would give rise to a difficulty, of the means of 

 surmounting which Mr' Talbot has given no idea. If x and v 

 are mixed up in f(xv), it is manifest that we cannot integrate 

 f(xv) dx, since v is a function of x, which if we eliminate we 

 merely return to our function X. We must therefore express 

 *f(xv) dx in the form Vdv, V being a function and, as Abel 

 has shown, an integrable function of v. Abel has given for- 

 mulas by means of which this reduction may be effected in all 

 possible cases. But there is nothing analogous to this in the 

 writings of Mr Talbot, and consequently he could not, setting 

 aside the defect already noticed, obtain results as general as many 

 previously known. In Mr Talbot's investigations, f(xv) dx is 

 such that ^f(xv) dx may be put in the form 



F X 2 {faxdx} + F 2 S W z xdx] + &c., 



<j> L Xj $ 2 x, &c. (of which Q'jX, (j>' 2 x, &c. are the derived functions) 

 being rational functions of x. Then ^(frx = a rational function 

 of v by a well-known theorem. Let the form of this function be 

 ascertained, and let us denote it by %v. Then differentiating, 



/ 

 and hence 



+ F 2% > + ...] dv, 



