if ON THE RECENT PROGRESS OF ANALYSIS. 65 
(an integer). Hence in multiplying an integral, the multiplier must be an 
integer, if y is rational in x, except for particular values of c. 
_ In the paper of which we are speaking Abel has applied precisely similar 
considerations to the case in which z and y are connected by any algebraical 
equation. 
__ Passing over one or two shorter papers, one of which has been already 
referred to at p. 59, we come to a ‘Précis’ of the theory of elliptic func- 
tions, published in the fourth volume of Crelle’s Journal, p. 236. The 
i work of which it was designed to be an extract was never written, and the 
_ §Précis’ itself is left unfinished. A general summary was prefixed to it, from 
which we learn that the work was to be divided into two parts. In the first 
_ elliptic integrals are considered irrespectively of the limits of integration, and 
_ their moduli may have any values, real or imaginary. Abel proposes the 
~ general problem of determining all the cases in which a linear relation may 
_ exist among elliptic integrals and logarithmic and algebraical functions in 
_ yirtue of algebraical relations existing among the variables*. 
__ His first step is to apply his general method for the comparison of trans- 
 cendents to elliptic integrals, which may be done by what is called Abel’s 
_ theorem, in at least two different ways: the one, that of which he now makes 
use; the other, that which we have seen is applied to the case of four func- 
_ tions by Legendre in his third Supplement. 
_ He next determines the most general form of which the integral of an al- 
_ gebraical differential expression of any number of variables is capable, pro- ° 
Piet it ean be expressed linearly by elliptic integrals and logarithmic and 
algebraical functions. The result at which he arrives admits of many im- 
portant applications. It is, that the integral in question may be expressed in 
a form in which the sine of the amplitude of each elliptic integral and the 
' corresponding A, and also the algebraical and each logarithmic function are 
all rational functions of the variables and of the differential coefficients of the 
integral with respect to each. 
He proceeds by an interesting train of reasoning to establish the remark- 
able conclusion, that the general problem which we are considering may 
ultimately be reduced to that of the transformation of elliptic integrals of the 
first kind. The problem of this transformation is then discussed, and by a 
method essentially the same as that of which he had made use in his paper in 
‘Schumacher’s Journal. The appearance however of the two investigations is 
dissimilar, because no reference is made to elliptic functions (as distinguished 
from elliptic integrals) in the first part of the ‘ Précis.’ The relations there- 
fore which exist among the roots of y=wW=2 are established by considerations 
’ independent of the periodicity of elliptic functions; though it is not difficult 
a 
x 
to perceive that they were suggested by the results previously obtained by 
means of that fundamental property. It is shown, that if the equation 
y= x, where yz is a rational function, satisfy the differential equation (A.), 
hen this equation, considered as determining x in terms of y, is always alge- 
raically soluble. As the multiplication of elliptic integrals may be consi- 
ered a case of transformation (that, namely, in which the modulus of the 
nsformed integral remains unchanged), this theorem may be looked on as 
an extension of that which we have spoken of (p. 58) in giving an account of 
Abel’s first memoir on elliptic functions. The two theorems are proved by 
he same kind of reasoning. 
The second part of the memoir was to have related to cases in which the 
moduli are real and less than unity ; of this however only the summary exists. 
f i In the assumed relation, the amplitude, or rather the sine of the amplitude of each 
‘ elliptic integral, is to be one of the variables, and noé¢ a function of one or more of them. 
