84 
12. The reduction of the quartid integral, usually called 
elliptical integral, to the cubic integral is remarkably easy. 
To effect this let the quartic integral be as follows : 
dy 
u 
=f“ = 
s/u- 
*h v (e - y)(f - y)(g - y)(h - y) 
Where, e, /, g, h are in the order of magnitude. 
Transform the integral (25) by the relation 
p + ex » — eq 
y = — =e+ l 
(25) 
q + X 
q + x 
,(26) 
From (26) dy = ^77-^3 • dx ; e - y = e< * ^ 
(q + x) 
7 e - h(hq -p \ 
h-y- ( — — 7 - x J 
q + x\ e-h ) 
q + x 
gq-p_ 
e-g 
xj 
Substitute these values in (25), observing the limits, then 
*ciq~p (27) 
dx 
eq-p 
q~.f+-g){e-h) 
e — a 
( h-y 
ftq-y* V e-/ 
J e- ft 
This integral only requires that eq>p in order to obtain 
its arithmetical value. 
If p = 0, then (27) becomes. 
f a q ( 28 ) 
dx 
I eg 
'“-\(e-f)(e-c 
(« -f)(e-ff)( e ~ h ) 
e — u 
ftq 
\TK7 X YjEIYSftTft 
<\e-f 7U-? } 
e — ft 
which is a cubic integral whose roots can be compared with 
(1). If the roots g h of eq. (25) are impossible, then 
equation (28) becomes a particular case of the equation 
considered in art. (9). 
If all the roots in eq. 25 are impossible, then it can be 
readily reduced to the form 
u — 
a 
dy 
J /V (l + y)( 1 + «V) 
(29) 
