210 MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 



To trace the curve BoP : — 



When ^ = zero, OBo = 



{i+n)-' 



when 6* = - =BoOB„ 0B,= '^^ 



when 6>= — = BoOB,, 0B,= -^ 



m {i+n)-' 



when6' = ^=BoOB„ 0B- = 



^2 



&c. &c. 



It is readily seen that the polar radii from OBj. to OB^. 

 are exactly the same as the polar radii included in the 



angle B^OBi, because cos ml \-(p] = cosni(p. 



The polar radii on each side of. OB, are equal at the 



same angular amplitude, because cos m|<pH — )=cos 



m( <pj ; therefore the polar area BqOB, is equal to the 



polar area B,OBi. 



From the above considerations the principle of the pe- 

 riodicity of cubic integrals is readily perceived, and it 

 follows that 



Cm dd _ Cm, 



Jo V I -^n COB m9 Jo '/i 



+ n cos mO' 



where p is any whole number. If the functional notation 

 be used, then 



^/(«'^.)=/("'f)-. • • • • (-) 



