EQUATIONS OF THE FIEST OEDEE. 
453 
(including the original equations) will be complete when no new equation arises from 
the combination of any one of the equations with any one of the equations of the 
original system'. 
I wiU illustrate this by investigating the conditions of integrability of the expression 
dx> ••• dx^)^^' 
If this expression admit of an integral V, it is easy to see that V will satisfy the two 
partial differential equations 
dx dy ~^y^dy^ *** '^y^' dyn-\ 
in which 
and <p stands for 
dyn 
= 0 , 
(2-) 
(3.) 
dy di^y 
dx' y^ dx^ 
y. ... yn)- 
The above are, in fact, the partial differential equations which we should obtain by 
Prop. I. as the equivalents of the system of ordinary differential equations, 
dY=<pdx, 
dy=y,dx, dy.—y^x, ... dy„_,=yjx. 
If we write 
{±\_d d d d 
\dx}-dx-^y^dy'^y^d^, - 
the above partial differential equations become 
(|)v=?...(i.), ^=o...(n.) 
The combination of (1.) with (II.) (by the theorem (1.)), then of (I.) with the result, and 
so on, gives a series of equations which may be thus expressed : — 
dY 
dY 
dyn--. 
dY 
* • 
= A.<p, 
in which 
dVo 
0 =A„<p, 
(III.) 
(IV.) 
(V.) 
(VI.) 
’'~dy~\dx)dyr+i'\dx)dy^+Y~ 
A ^ / d\ d , / d\^ d 
“ dy \dxjdy^ '\flxj dy^ 
3 Q 
MDCCCLXII. 
