MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 211 



Froin the figure it will be seen that 



where <p is less than — . 



^ m 



1 1 . Approximate formula for calculating the cubic in- 

 tegral : — 



Put dj = — , /•(,, r„ r^, &c., the polar radii corresponding 



to the angles zero, — , — , &c. 

 pm pm 



I 



It can be readily shown tliat 



n 



o \^i +n cos niQ 



where p is any whole number. 



This formula is obtained by supposing the curve BqP 

 to coincide with the curve whose equation is r'^ = ao + a^& 

 + aj^. The values of r^, r,, &c. are as follows : — 



-v/j +n' 





a/ i+wcos^ ^ i-\-ncos^ 



v2 



