ON THE RECENT PROGRESS OF ANALYSIS. 63 
may be satisfied. [I have altered his notation for the sake of uniformity. ] 
» Wx be the function sought, then considering y = x as an equation de- 
termining x in terms of y, he shows that certain relations necessarily exist 
among its roots. Let A be one of them and 6! another, it will readily be 
seen that we may put ' 
. di'!=d6, 
_ since each is equal to 
- a 
Vy) U-#y’) 
ih 6! = 6 + a, 
Al! 
ie @ being the constant of integration, or, which is the same thing, being inde- 
pendent of y. Hence a4 being one root, every other root is necessarily of 
Rm the form A(§+ a). Again, we see from hence that 
] y=) = Yat a)), 
iH 
a which is to be true for all values of 8, and which therefore implies the exist- 
ence of a series of equations, of which the type is 
A 
i VAG+H—1a)) = (a+ ha)), 
§ where # is an integer. Hence A(9 + &«@) is a root, whatever integral value 
ba we may give tok. But the equation y= wz has but a finite number of 
_ roots, and therefore the values of the general expression A(§ +a) must 
 Yecur 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 
_ eompound) of the function A 6. 
__ All the values of A (@ + ka) got by giving different values to & 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 ina more general expression, viz. A(9+h, a, +h. @.-k, &,), 
and conversely, that all the values of this expression are roots. The number 
_ mis indeterminate: we may have formule of the form y = Ww 2, in which 2 
__ is unity, others in which it is two, &c.; but in all cases a is an aliquot part 
f some period of A 6, and & is integral. 
It is easy when the roots of y =z aré known, to express y in terms of 6. 
wr let Pz = a J and F being integral functions. Then 
yFe—fe=(yp—q {(@—Ab) (@— AO +.4)) oe} 
is (yp —q being the coefficient of the highest power of « inyFa—fw) an 
nti¢ally true equation; whence, to determine y in 9, we have only to assign 
a particular value to 2, or to compare the coefficients of similar powers of it*. 
This then determines the form which the function y must necessarily be 
: the question which Abel goes on to discuss is this: Under what circum- 
nees will a function of the form thus determined @ priori be such a func 
mn as we require? The character of the reasoning by which this question 
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 ¢, denote the function inverse 
ag 
_* T have trot noticed an ambiguity of sign at the outset of this reasoning, as given by 
Abel, as for the purposes of illustration it is immaterial. 
