79 
The limits become 
when x — a'.z — 
x — pt.'.z — 
2a — a — b 
a-b 
2/3 - a - b 
a-b 
Substitute these values in the cubic integral (u), then 
f2a - a-b 
clz 
u 
V. 
a + b - 2c 
a-b 
{») 
J a-b 
6. Transformation of the cubic integral (8) by the rela- 
tion 
2 = cos.m6> (9) 
where (m) is any whole positive number. 
From (8) dz=—m$m.mOdO=:—-m\/l—z 2 .dO f then (8) be- 
comes 
U = 7)1 
j 
cos. 1 
m 
a + b-2c 
(2fi - a-b 
\ a-b 
2a- a-b 
dd 
1 + a ^ . cos.m0 
a + b~2c 
1 , r2a-a-b\ I a-b a ( 10 ) 
0OS — A [ 1 * / 1 » nnfl <yv)H 
m ’ V a-b 
Let u 2 represent the integral (10) between the limits 
a— a and (3=b, then 
Uo = 
a 
dx 
b yj (a- x)(b - x)(c - x) 
= Y 
a + b - 2c 
7 r 
m 
0 
j 
dd 
- a-b 
1 + ; — ?rcos.m0 
a + b -2c 
(ii) 
By comparing (6) and (11) we obtain 
a 
dx 
dx 
. =—...( 12 ) 
i b \J (a - x)(b - x)(c - x) ^ _ o0 \/ (a - x)(b - x(c - x) 
This result is neat and gives an answer to question 5508 
Educational Times for December, 1877. 
7. Since 2a<2$, then 2a— a — 5<a — 6; therefore 2a — a — b 
upon a — b and 2/3 — a — b upon a — b are always proper 
