78 
The value of ^/a+'b - 2 c is real because a + b - 2c is positive. 
The fraction a - b upon a + b-2c lies between 1 and zero, 
it is zero when a = b and 1 when b = c. At each of these 
limits the cubic integral is soluble. 
4. Transformation of the cubic integral (3) by the rela- 
tion 
-2 = cos.m0 (4) 
Where (m) is any whole positive number. 
From (4) dz = m&m.mddQ = 1 - z 2 .dO 
Substitute these values in (3) and it becomes 
1 -1/2 c-a-a\ 
dd 
u 
— m / 
V 
2 
a + b ~ 2c 
-cos 
m 
1 /2c -a- a\ 
\ a — a ) 
1 -1/2 c-a-j3 
j 
, a - b 
i + ; — — .cos.mft 
0) 
a + b - 2 c 
- cos 
J m 
/2c- a- /3\ 
V / 
Let n x represent the integral (5) between the limits a = c 
and (3 = - oo , then 
a c 
u x — 
dx 
>J {cl- x){b - x){c - x ) 
oo 
r 7r 
m 
/ 2 
-m . f 
V a + b - 2c 
dd 
(6) 
J 
J 
a - 6 
1 + ^ — ^-cos .md 
a±b-2a 
0 
5. The integral in equation (5) applies only to case (1) in 
Art. 2 ; the second case requires a slightly different trans- 
formation, as follows 
Transform the cubic integral (u) into another cubic 
integral by the relation 
a + b ( a-b)z 
x = 
+ 
(0 
_ , a - b _ a - & . 
From ( 7 ), dx = — ^ - .dz, a - x = — (1 — ») 
-{b-x) = -s- (1 + *), - (c - a?) = 
( 
1 + 
(a - 6)2 
a + 6 - 2c 
) 
