10. The comparison of the cubic elementary integral, viz. 
\e dd ( 19) 
JO v'l + ncos.mO 
with elliptic functions of the first order will be considered 
hereafter. It will be necessary now to show the march of 
the integral (19) with respect to polar coordinates in the 
usual way. 
Let B 0 P be the polar curve traced by the equation 
V 2 t 
1 1 + ncos.mO j. 4 
where r— OP and 0— B 0 OP. 
To trace the curve B 0 P 
When, 6 = zero 
„ 0 = — = BoOBi..., 
m 
ii 
r-i 
cq 
O 
„ B - 2r -B 0 OB 2 ... 
m 
....'.OB 2 = 
„ 0 = ~ = B o OB 3 .... 
m 
, . .'. OB 3 = 
&c. 
&c. 
(1 +n)* 
(1 - n ) 4 
(1 
(1 -nY 
It is readily seen that the polar radii from OB 2 to OB 4 are 
exactly the same as the polar radii included in the angle 
( 2 r \ ° 
— + (/>) = cos .mcj). 
The polar radii on each side of OB 2 are equal at the same 
angular amplitude, because cos.m (^[> + ^ = Cos.mQ^ - ^ 
Therefore, the polar area ByOBi is equal to the polar area 
B, OB 2 . From the above considerations the principle of the 
