EQUATIONS OE THE EIEST OEDER. 
-44B 
The proof of this will consist of two parts : — 1st. If will be shown that, if a systeni of 
p integrals exists, the conditions represented by (8.) will be identically satisfied. 
2ndly. It will be shown that, when the conditions represented by (5.) are identically 
satisfied, the solution either of the final system of partial difierential equations (4.), or 
of the corresponding system of ordinary difierential equations, by a system of^ integrals 
is theoretically possible. 
It will follow from these conjoined, that the number of actually existing integrals is 
exactly p. 
1st. The system of ordinary difierential equations corresponding to (4.) may be ex- 
pressed in the form 
( 6 .) 
— + IIp2^'^p+2 ••• Hpm^^p+m* J 
Now suppose this system to have p integrals. Then, by means of these, ^i, ... Xp can 
be eliminated from the coefiicients II„, &c. in the second members, which will thus 
become exact difierentials of those functions of the variables Xp+^ ... Xp^,^ which express 
the values of x^, ... Xp. Hence we shall have the system of conditions 
d 
dxp+i 
dx 
■H... 
p+; 
ki') 
( 7 .) 
Jc representing any integer from 1 to p, and j any integers from 1 to m, and the 
bracketed symbols of difierentiation referring to H*^-, as transformed. Hence, the 
unbracketed symbols referring to the prior state of the functions, we have 
dx-^ d 
dx, 
p 
dxp+i'dxp+i dX]^ ■’* ' dxp+i dXp 
In the same way 
so that (7.) becomes 
( 8 .) 
Now if we construct, in analogy with (4.), Prop. H., the developed form of the con- 
ditional equation (5.) of the present section, we shall have 
(A,H,- A,H.4| . . . +(A.H„.- A,H„) §=0, 
or, 2 denoting summation fi’om k—l to k=p, 
2(Aft,-A,.H„)f =0.^ , 
Now as by (8.) the coefficients vanish identically, the equation, and generally the system 
of conditional equations of which it is the type, will be identically satisfied. 
