the calculus of functions. 239 
and also differentiate the results. Then we shall have the 
two following sets of equations 
(«) 
4/** = F [x, iK-j} 
( s ) 
&c. 
&c. 
4// -1 j = F \ J- Z x, i/ot- z x, ' 1 (/’—I) 
1 1 dx \ dx J J 
t I 
X — F | as P ~~ l JC y \\jpf~~ x , | ( p ) 
4^ ^ — 
JC£ 
and also 
= iipu ^if 7 
^ I ^ 77 { 
d-\/ot, x d. f ■ dipux 
- _ 
dx 
&C. 
_cL 
dx 
d&x 
dx 1 dx 
TJ 
(«> 1 ) 
(^2) 
&c. 
— ii F I oc p ~"fJ~ 2 x, [ da - P } (*,/>— 1) 
</# dx t \ dx \ dx j J 
* 
- A. f If f j | (a,/>) 
di/aP X 
dx 
Since u^x = x equation (p) becomes 
tyx = F > a. P~~ 1 X, ityc&p l x, X l^ d*P x | | 
from this by means of equations [p — j) and (a,p — 1) we may 
eliminate and^i^-—f the resulting equation will con- 
tain only x> fyx, t\jccx, &c. -^aP—zx and their differentials. From 
this by means of (/>— 2) and {a,p—<i) we may eliminate ^ap- z x, 
and its differential, leaving an equation containing only x,\px,\pax, 
&c. §aP~~ 3 x and their differentials. In the same manner 
*x may be eliminated, and the process may be continued 
