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value of the positive ordinate. The positive branch extends 
only as far as A, where OA = a. All values of the ordinate 
( y ) beyond this point are impossible. The ordinate (y) is 
impossible also between B and C, where OB = b and OC = c, 
since the denominator of (1) is then negative. 
When, x — a = OA 
x = b — OB 
x = c — OC 
x<ia and ">b 
x<d> and >c 
x<c 
. .y = a> 
‘.y = oQ 
:.y = oo 
‘.y is real and positive. 
.\y is impossible. 
y is real and positive. 
In this geometrical representation of the cubic integral 
there are two cases to consider, viz. : 
1. When the limits a and j3 are both less than c. 
2. When the limits a and j3 lie between A and B. 
In case (1) the cubic integral is correctly represented by 
the area HDEK, where OE = a and OD = j3. 
In case (2) the cubic integral is geometrically expressed 
by the area LFGM, where OG = a and OF = j3. 
3. Transformation of the cubic integral (u) into another 
cubic integral by relation— 
2c 
_ a + az_ 2 (a-c) 
x — — — ct ■ — - — — • 
1+0 
l+0 
From (2), dx = — Q.dz 
v n (1 + zf 
a + b -2c 
a - x 
_ 2 (a - c) 
1+0 
b - x = 
1+0 
/ 1- (a-fy* \ 
l a + b - 2c j 
x — 
a-c 
1 +0 
■(2) 
(1-0) 
The limits become as follows 
When, x 
a 
x = 
.0 = 
.0 = 
a + a - 2c 
a — a 
a + ^3 — 2c 
a - (3 
Substitute these values in the cubic integral (u) and it 
becomes 
fa + a - 2c 
“=V 
a + b - 2c 
a - a 
a + j3 - 2c 
dz 
a 
p 
(i) 
