Euler's addition theorem for elliptic integrals states that, with 
,
 |
(1)
|
where
 |
(2)
|
For real variables with 
and 
, use positive square roots and require the sum of the two integrals
to lie between 
and 
, where 
is the complete
elliptic integral of the first kind. The formula follows from the addition identity
for the Jacobi elliptic function 
(DLMF 22.8.1).
A form of Euler's differential equation is the separable differential equation
 |
(3)
|
where 
is a polynomial of degree at most four (Euler 1761,
§30). On real intervals where 
is positive, take positive square
roots. If 
, separation gives 
for a constant 
. The addition theorem above gives this sum as 
when 
, 
, 
, and 
.
."
Novi Comment. Acad. Sci. Imp. Petropol. 6, 37-57, 1761.