202 



MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 



where w is a proper fraction, and m any positive whole 

 number. I believe this elementary integral is better 

 adapted to the purpose of tabulation than is the elemen- 

 tary integral used by Legendre and others, viz. 



dO 



r 



Vi-A^sin*^' 



to which the former integral can be readily reduced. In 

 tracing the curve whose polar equation is r*=2x 

 V I +ncosm6 by polar coordinates r, 0, the periodicity of 

 cubic integrals is readily perceived. 



And here I must express my surprise that this method 

 of tracing the cubic integral has not been made use of in 

 tracing elliptic fimctions. That this has not been done I 

 infer from the circumstance of not finding it in Cayley's 

 recent Treatise on Elliptic Functions. 



2. Let a, b, c be positive, and taken in the order of mag- 

 nitude, that is a>b>c. 



If the roots are not positive, they can be readily made so 

 bv the linear transformation x=K-\-z^. 



Fig. I. 



With a view, therefore, of fixing palpably the values of 

 the limits a, /3, with respect to the roots o, b, c, it will be 



