454 ON SIMULTANEOUS DIEEEBENTIAL EQUATIONS OE THE EIEST OEDEE. 
The combination of (IT.) with (HI.) ... (V.) gives the series of conditions 
A 
dyn 
Ai<p=0, 
(vn.) 
The conditions of integrability are expressed by (VI.) and (VII.). These satisfied, the 
equations (HI.), (IV.), ... (V.) show that <^dx can be expressed as an exact differential 
with respect to x, y, y„_i, in the form 
<p(Z^z=((p— yjAj<p— y2A2(p ... —y^A„<p)dx 
+A,<pd^+A,(pdtj^ ... +A,<pd^„_„ 
a result first established by M. Saeeus. 
