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222 
EXACT DIFFERENTIAL EQUATIONS. 
For the solution of an exact differential equation, it is there 
fore only needful to equate to c the integral of the correspond 
ing exact differential as found by the above process. 
The failure of that process, through the occurrence of a 
form in which the highest differential coefficient is not of 
the first degree, indicates that the proposed function or equa 
tion is not ‘ exact.’ 
5. There is another mode of proceeding of which it is pro 
per that a brief account should be given. 
d 2 y d n y , . . .. 
Jx^'"dx n ' 11 1S easi V 
shewn by the Calculus of Variations, that if Vdx be an exact 
differential, V being a function of x, y, y x ,... y n , then identically 
Representing ^, 
\dV, 
f d y 
VdV 
_ fd\ 
VdV 
V dy x 
\dxj 
1 d Vz" 
\dxj 
1 dy n 
= 0 (37), 
wtere (I) 
indicates that we differentiate with respect to x 
regarding y, y n as functions of x. This condition was 
discovered by Euler. 
The researches of Sarrus and De Morgan, not based upon 
the employment of the Calculus of Variations, have shewn, 
1st, that the above condition is not only necessary but sufficient. 
2ndly, that it constitutes the last of a series of theorems which 
enable us, when the above condition is satisfied, to reduce 
Vdx to an exact differential in form, i. e. to express it in 
the form 
dU , dU , dU, dU , 
H dx+ -dj d y + dj l d!,l "" + dyZ !/ '- i (38) ’ 
where x, y, y l ,...y n _ l are regarded as independent. The inte 
gration of Vdx — 0 in the form U= c is thus reduced to the 
integration of an exact differential of a function of n + 1 inde 
pendent variables,—a subject to be discussed in Chapter xii. 
(Cambridge Transactions, Vol. IX.) 
The condition (37) is singly equivalent to the system of 
conditions implied in the process of Sarrus. The proof of this 
equivalence a posteriori would, as Bertrand has observed, be 
complicated. (Liouville, Tom. xiv.)