ON THE RECENT PROGRESS OF ANALYSIS. 283 



implies the existence of a series of equations, of which the 

 type is 



where k is an integer. Hence X (6 + ka) is a root, whatever in- 

 tegral value we may give to k. But the equation y tyx has 

 but a finite number of roots, and therefore the values of the 

 general expression X (6 + Tea.) must recur again and again. This 

 consideration throws light on the nature of the quantity a ; it 

 must in all cases be an aliquot part of a period (simple or com- 

 pound) of the function \0. ^ 



All the values of X (9 + lea.} got by giving different values 

 to Jc are roots ; but the converse is not necessarily true ; all the 

 roots are not necessarily included in this expression. But it is 

 not difficult to perceive that all the roots are included in a more 

 general expression, viz. X (6 +Jc^ a l + & 2 2 . . . & n ot n ), and conversely, 

 tli at all the values of this expression are roots. The number 

 n is indeterminate : we may have formulae of the form y = tyx, 

 in which n is unity, others in which it is two, &c. ; but in all 

 cases a is an aliquot part of some period of X0, and It is integral. 



It is easy when the roots of y tyx are known, to express y 

 in terms of 0. For let 



fx 

 tyx =~,f and F being integral functions. Then 



is (yp q being the coefficient of the highest power of x in 

 yFxfx) an identically true equation; whence, to determine 

 y in #, we have only to assign a particular value to a?, or to com- 

 pare the coefficients of similar powers of it*. 



This then determines the form which the function y must 

 necessarily be of: the question which Abel goes on to discuss is 

 this : Under what circumstances will a function of the form thus 

 determined h priori be such a function as we require? The 

 character of the reasoning by which this question is treated is 

 similar to that of the method by which Abel had, in his second 

 memoir on elliptic functions, verified the form which, without 

 assigning any reason, he had there assumed for the function y. 



The second essay is singularly elegant. If <f> k denote the 



* I have not noticed an ambiguity of sign at the outset of thia reasoning", 

 as given by Abel, as for the purposes of illustration it is immaterial. 



