of certain differential Expressions , &c. 
%35 
Legendre’s series is easily reducible to this, since log. = 
i log- x = lo g- 2 — 2 log- b. 
This memoir of Euler ( Animadversiones in Rectificationem 
Ellipseos ) is curious, on account of the strange artifices used to 
obtain the series for the length of the eccentric ellipse. It is cha- 
racteristical of the peculiar mathematical powers of Euler, and 
also bears strong marks of the rapidity and eagerness with which 
he conducted every work of calculation. The author discovers the 
series and its law, partly by tentative methods, and partly by the 
use of a differential equation of the second order; and indeed, 
without the use of such an equation, it is difficult to exhibit the 
law. Let/(i) represent the whole integral of /, from x = o to 
x = 1 , then, 
(l-fr*) .rfV(l) __ 1 +b z df( I) , r, > _ 
db? b * db 1“ J ( 1 ; — 
Assume then, /(i)= 1 + A6 2 + B6 4 -j- Cb c -f &c. 
-j-|« 6 E + /3 b- + yb 6 4 - &c. Jlog. b ; 
deduce the values of - ^ compare the terms affected with 
like powers of b ; and the law of the series, such as it has been ex- 
hibited, may be deduced. 
The following is the method of deducing the differential equation ; 
df=dzj •••-!- = — Sr* and > takin S the P artia! 
differentials, 
d*f bz* 
- (i+z 2 b 
dx ' db V[i+ *?)$+*& 
consequently, = 
and 
drf 
dz 
— bz 1 
V(I + & 1 * 2 ) (i+z 2 H 
d'f 
b d*f 
dz . db' 
, b.d*f 
dz . db 
b * z a 
V(I + &z*)i (i+itli 
_V _ £/. 
(x+6 4 zql(r+z l ]f dz.db*’ 
dz.db « dz.dhr- 
