THAN THE FIRST, CONTINUED. 
211 
....(9), 
(10). 
ss z in terms 
integrate 
(ii). 
, we cannot 
may proceed 
(12), 
the right-hand member indicating the performance of n — 1 
successive integrations, each of which introduces an arbitrary 
constant. If between this equation and (10) we, after integration, 
eliminate z, we shall obtain a final relation between y, x, and 
n arbitrary constants, which will be the integral sought. 
Ex. 
Given 
d 2 y d s y 
TV-T oVun rv d y. 
» = c + av'(l+2 s ) (a). 
According to the first of the above methods, we should now 
solve this with respect to z, and thus obtaining 
= / 
dx 2 V 
x — c 
a 
-1 
find hence 
V 
x — c 
1 r dx^ + c L x + c 2 
(&), 
to which it only remains to effect the integrations. According 
to the second method, we should proceed thus. Since 
, azdz . 
dx = —tt: t—or, we have 
VU+s 2 ) 
az*dz 
V(1 + z 2 ) 
azdil+z 2 ) r as, , 
= — L - 5 log {z+ V(1 + s’)) + e , 
azdz 
V(i + 
for dx, 
whence multiplying the second member by 
and again integrating, 
y-~jV(i + <)log{*4V(i+^} + ff* 
+ ac V (1 + z 2 ) +c" (c). 
The complete primitive now results from the elimination 
of z between (a) and (c). 
14—2