f 



MR. ROBERT RAWSON ON THE CUBIC INTEGRAL. 207 



Hence, in both cases refered to in art. 2, the following 

 ntegral will apply : — 



, a—b ^ 2c—a—a 28 — a — b 



where n = y— — , cos^,= or -^- , 



a + b — 2c a-u a—b 



J a 2c—a — B 2u — a — b 



and cos a. = ~- or = — . 



a— jS a—b 



8. It appears, therefore, that if the elementary integral 



=====r (14) 



Vi +WCOS md 

 is calculated, for any given value of the modulus n, from 



^=zero to 6=—, the result will be a complete solution 



m ^ 



(arithmetical) of the cubic integral in the case when all the 

 roots are real. The following investigation will show that 

 the same elementary integral will apply equally well when 

 two of the roots are unreal. 



g. In the case of the cubic integral having only one 

 real root it assumes the form 



r dx . . 



v = \ =-. . . (15) 



J^ \/{a-x){{x-bY + c'] ^ ^' 



where a, b, c are independent constants. The limits u and 

 /9 cannot be greater than a. 



Transform the cubic integral (15) by the relation 



A—B cos — 



x = ^, .... (16) 



I — cos 



2 



where A = a- 'v/(a-6)* + c% B = a+ \/{a-by-\-c\ 



