83 
periodicity of cubic integrals is readily perceived, and it 
follows that 
> dd 
m 
V 
— dO 
m 
✓0 0yj . . 
Where, p is any whole number. If the functional notation 
be used, then 
Ef(n > —)=/( n > —) ( 20 ) 
\ m J \ m / 
From the figure it will be seen that 
/{»> (^ r+ *)} -/("» = 2 /( w > ( 21 > 
f{+ ? ) = 2/ ( n ’ m) (22) 
•'■/ { “> - i) } + 2 M *) = / { »> (~ + *) } -( 23 ) 
• 7T 
where 0 is less than 
T m 
11. Approximate formula for calculating the cubic 
integral— 
Put 0i — r 0 , r v r 2 , &c., the polar radii corresponding 
to the angles zero, — , — , &c. 
0 jpm 
It can be readily shown, that 
(24) 
J f ==K = T- U + + 4(n 2 + *4 + • .»w + 8(f4 + . . . r;_ 2 ) } 
vl+wcos.m0 Gm^>( J 
JO 
where (p) is any whole number. 
This formula is obtained by supposing the curve B 0 P to 
coincide with the curve whose equation is r 2 = a 0 + a-fi + <x 2 0 2 . 
The values of r 0 , r v &c., are as follows 
, 2 
V + ncos.md 
0 
1 + wcos .m0 
r 
r — 
\/l + n 
2 
yj 1 + ?IC0S. 
r — 
V1+ wcos.* 
V 
~ == 
3 \/l+7lCOS. 3 ^ 
Z> 
1 
\/ 1 + 
%COS.~ 
r' 2 = 
\/l — 
The larger p is the more accurate is formula (24). 
