MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 313 



If jo=o, then (27) becomes 



eq 



{e-f){e-ff){e-hy 



e—a 



^■^ (28) 



which is a cubic integral whose roots can be compared 

 with (i) . If the roots ^ and h of equation (25) are impos- 

 sible, then equation (28) becomes a particular case of the 

 equation considered in art. 9. 



If all the roots in equation (25) are impossible, then it 

 can be readily reduced to the form 



„r j!^_^ _._, . . . (,,) 



J, V(i +!/')(■ +«y) ^' 



Transform (29) by the relation 



- I — cos mQ , , 



^ I + cos m^ ' ^^ ' 





.'. u = 



A / I H , cos m9 



. . (31; 



\/2(-fa*) 



■ '% / 1-1 — 



The equation (31) is therefore of the form given iu 

 art. 7. 



