MR. ROBERT RAWSON OX THE CUBIC INTEGRAL. 



209 



where 



\^{a — by + c^ + a — b 

 'S ^{a-by + c^ + b-a 



cos 



0,_a-u- s/{a-bY + c^ 



e «-/3- sJ{a-bY + c'- 



and cos-= . 



2 a-/3+ 'sJ{a~bf + &- 



10, The comparison of the cubic elementary integral, viz 



d0 



i s/i 



V I + w cos md' 



(19) 



with the elliptic functions of the first order will be consi- 

 dered hereafter. It will be necessary now to show the 

 march of the integral ( 1 9) with respect to polar coordinates 

 in the usual way. 



Fig. 2. 



Let BqP (fig. 2) be the polar curve traced by the equation 



._ ^/2 



{i +»cos md]^ 



where r = OP, and^=B,OP. 



SER. in. VOL. VI, ^ 



