[656 
657] 
139 
NTS. 
! o> 
657. 
determine also 
; a constant, we NOTE ON THE THEORY OF ELLIPTIC INTEGRALS. 
ial ; I have not 
[From the Mathematische Annalen, t. xii. (1877), pp. 143—146.] 
The equation 
Mdy _ dx 
Vl — y 2 .1 — k 2 y 2 Vl — x 2 .1 — k 2 x 2 
is integrable algebraically when M is rational : and so long as the modulus is arbitrary, 
then conversely, in order that the equation may be integrable algebraically, M must 
be rational. For particular values however of the modulus, the equation is integrable 
algebraically for values of the form M, or (what is the same thing) , = a rational 
quantity + square root of a negative rational quantity, say =^(l + mV-n), where 
l, m, n, p are integral and n is positive ; we may for shortness call this a half- 
rational numerical value. The theory is considered by Abel in two Memoirs in the 
Astr. Nach. Nos. 138 & 147 (1828), being the Memoirs* XIII & XIV in the Œuvres 
Completes (Christiania 1839). I here reproduce the investigation in a somewhat altered 
(and, as it appears to me, improved) form. 
Putting the two differentials each = du, we have x — sn(u + a), y = sn ; and 
the question is whether there exists an algebraical relation between these functions, 
or, what is the same thing, an algebraical relation between the functions x = sn u and 
u 
y=SU M‘ 
Suppose that A and B are independent periods of sn u\ so that sn (u+ A) = sn u, 
sn (u + B) = sn u, and that every other period is = mi + nB, where m and n are 
integers. Then if u has successively the values u, u + A, u + ZA, etc., the value of x 
[* They are the Memoirs xix. and xx. in the Œuvres Completes, t. i., Christiania, 1881.] 
18—2