208 MR. ROBERT RAWSON ON THE CUBIC INTEGRAL, 



From ( 1 6), dx=m \/ [a — by + c^ 



_V i-cos — 



"7 ^^'^^' 



(_i-cos-j 



1 + cos 



me 



a-x— \/{a-by + c^ 



I — cos 



m0' 



,,. . „._ x/{a-by + cH3 s/{a-br + c^ + b-a} 

 (^i-cos-j 



{ 



1 + 



( \/{a — by + c'- + a — b)cosmd 



i — b)cos m6') 

 ^ + b-a J 



3 \/{a-by + c'-^b- 



Substitute these values in (15), observing the change in 

 the limits, then 



v = 



m 



V3 \^{a—by + c^ + b-a 





+ V(a-6)2+cz'' 



dO 



^ / \/{a-by + c' + a-b . 



\/ i-\ — ,- : COS mU 



^ 3 V(«-6)" + c^ + 6-a 



[17) 



It is readily seen that 3 V (« — ^)^ 4 c'' + ^ — a is a positive 

 quantity, whatever may be the values of a, b, c. The co- 

 efficient of cos md is a proper fraction ; therefore equation 

 (17) may be written as follows : — 



v= 



\/3 V(«-6)' + c* + 6 



_____ t "" 



jW 



+ « cos m$ 



„ . (18) 



