of certain differential Expressions, &c. 
253 
for/.y.y.put 
equation between /(1 ), /( 1 ), '/ ( 1), the same as the one given in 
the preceding page. 
Landen, Mem. page 35, and Legendre, Mem. de l’ Acad. 178 6 ’ 
p.678, have deduced an equation subsisting between the circum- 
ference of a circle and the peripheries of two ellipses, whose 
excentricities are and dilfdL;* but the application of the 
I^|V2 
preceding forms will enable us to express, immediately, the rela- 
tion between the peripheries of a circle and of two ellipses, the 
excentricity of one ellipse being assumed of any magnitude; 
thus, by equation («), page 242. 
And /, j', may represent arcs of ellipses described on the same 
semiaxis, major ( 1 ), with excentricities equal to e, e\ being 
__ 1 — \/(i— e a ) 
i + \/(i-eT 
or, ( 1 -e') . P . 7T— 4 /'( 1 )+2.( i+e') /( i)=o ; 
or, since -L = quadrant of circle ( q ) radius =1, 
(1— «') p -?- 2 /'( i )+(h-0 /( 1 )= 0 - 
* The semiaxes of the t\yo ellipses compared by Landen, are i, and — L_ 
