MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 205 



Let M, represent the integral (5) between the limits a= 

 c and /3= —00 , then 



z'" dx 



' J ^ {a — x){b — x){c—x) 



/ 2 r^ «?^ 



^ . (6) 



cos mO 

 , + b — 2a 



5. The integral in equation (5) applies only to case i 

 in art. 2 ; the second case requires a slightly different 

 transformation, as follows : — 



Transform the cubic integral u into another cubic in- 

 tegral by the relation 



a-\-b {a—b)z 



x= + ' 



2 2 



(7) 



From (7), da^= dz, 



a — b, > 

 a—x=^- -(I — z) 



2 ^ '■ 



, , , a — b , , 



^ ' 2 \ a-\-b — 2c/ 



The limits become : — 



when x-=u, 2; = ' 



2« — a — b 



a-b ' 



2^-a-b 



when x=p, z= 7 — • 



' a — b 



Substitute these values in the cubic integral u, then 



<za—a—b 



^^ = \/^ + ^ 



dz 



+ b — 2C 



\/('-')(-^iT^J 



■ (8) 



zp — a—b 



