80 
fractions whatever may be the values of the real positive 
roots a, b y c. 
The limits of both (5) and (10), integrals which repre- 
sent the cases 1 and 2 in Art. 2 respectively, are angles 6 lf 
and 6 2 which lie between zero and (jr). 
Hence, in both cases referred to in Art. 2, the following 
integral will apply : 
u 
=, V; 
fe L 
m 
dd 
Where n = 
a + b-2cQ_ 2 \/l+ ?icos.m0 
J m 
a-b 
,(13) 
a + b - 2c 
and cos0o 
„ 2c -a- a 28 - a-b 
cos.@i — or ' 
a — a 
a-b 
2c- a- 1 3 or - a-b 
a — f 3 a-b 
8. It appears, therefore, that if the elementary integral 
dd 
. ■ .... ..........(14) 
10 \/l + ncos.md 
is calculated, for any given value of the modulus (?i), from 
TT • • 
6 = zero to 8 = — 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. 
9. In the case of the cubic integral having only one real 
root it assumes the form 
fa dx ... 
V =d = (15) 
j/5 J(a-x){(x-by + c 2 } 
where a, b , c are independent constants. The limits a and 
(3 cannot be greater than (a). 
Transform the cubic integral (15) by the relation 
a r> 
A - Bcos.-y 
— - — 5* < 10) 
1 - COS.“ 
where, A •a*- J{a-bf + c‘S B = « 4 ‘v / (a -b)~+? 
