116 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
53. We consider the differential expression 
dV—pdx — qdy — rdz, 
which, treating the integrals as constants, that is, in the expressions of V, x, y, z, 
regarding t as the only variable, is at once seen to be =0; hence, if we regard all 
the integrals as variables, the value is 
= d\ 4- Ada + Bdb + Cdc + Ddd + Ede, 
without any term in dr, since this enters originally in the form dt — dr, and there 
fore disappears with dt. 
The coefficients A, B, G, D, E are of course functions of a, b, c, d, e, t — r; 
it is to be shown that they contain t — r linearly, viz. that in these coefficients 
respectively the coefficients of t — t are 
dJH dH dH dH dH 
da’ db ’ de’ dd ’ de’ 
where H is expressed as above in the form H (a, b, c, d, e) ; this being so, the entire 
term in t—T will be (t — r)dH; each coefficient, for instance A, has besides a part 
A', which is a function of a, b, c, d, e without t—r, or changing the notation and 
writing the unaccented letters to denote these parts of the original coefficients, the 
final result is 
dV — p dx — qdy — r dz — {t — r) dH + d\ + Ada + Bdb + Cdc + Ddd + Ede, 
where H stands for its value H (a, b, c, d, e), and A, B, C, D, E are functions of 
a, b, c, d, e without t—r. 
54. To prove the theorem, we have 
. dV dx dy dz 
da ^ da ^ da T da’ 
and thence 
dA _ d 2 V dp dx dq dy dr dz d?x d?y d 2 z 
dt dadt dt da dt da dt da ^ dadt ^ da dt 1 dadt 
d idV dx dy dz) 
da { dt P dt ^ dt T dt) 
^ dp dx ^ dq dy ^ dr dz dp dx dq dy dr dz 
da dt da dt da dt dt da dt da dt da ’ 
dV 
and then substituting for ^ , &c., their values from the system of differential equations, 
the first line vanishes, and the second line becomes 
_ dH dp ^ dH dq dH dr dH dx dH dy dH dz 
dp da dq da dr da dx da dy da dz da’ 
_dH 
da ’