128 A MEMOIR ON DIFFERENTIAL EQUATIONS. [655 
Forming from these the expression for dV—pdx —qdy — rdz, the term in dt — dr 
disappears; there is a term in t — r, the coefficient of which is 
3 d. abc — ad.bc — bd.ca — cd.ab, 
which is = d. abc, or the term is (t — r) dH; and we have, finally, 
d c 
dV — p dx — q dy — r dz = d\ + (t — t) dH — bdj—cd-\ 
0 c 
viz. t enters only in the combination (t — t) dH, which is the fundamental theorem. 
Considering if as a determinate constant, this term disappears. 
We may show how this formula leads to the solution of the partial differential 
equation pqr = 1 + H; treating H as a definite constant, then in order that the 
formula may give dV — pdx —qdy—r dz — d\, or V=\+J (p dx + q dy+r dz), as before, 
d e 
the last two terms of the formula must disappear; this will be the case if and - 
0 c 
are constants, or, say, d = b(3, e = c<y, /3 and y being constants. But, this being so, we 
have qfi = qy — px, r<y = rz —px, that is, px = q (y — /3) = r(z — y), pqr = 1 + H, giving the 
values of p, q, r; and then 
V=X+ j" (pdx + qdy + r dz), = A, + 3 ^/(1 + H) x* (y — ¡3)* (z — <y)\ 
which is substantially the same solution as is obtained above by a different process. 
Or, again, observing that we have 
dV — pdx — qdy — rdz = d\ + (t — r) dH — dd — de — ^db — ^dc, 
then, taking H, b, c constants, we have 
dV —p dx — qdy — r dz = d\ — dd — de, 
which, changing the value of A, gives the before-mentioned solution 
V = A + ax + by + cz, (abc = 1 + H). 
75. As a second example, suppose 
H=^ (p- + q 2 + r 2 — x 2 — y~ — z 2 ); 
the augmented system is 
dx _dy _dz _dp _dq _dr _ ^ _ dV 
p q r x y z p 2 + q 2 -hr 2 ’ 
corresponding to the dynamical problem of the motion of a particle acted upon by a 
repulsive central force equal to the distance. 
The integrals of the original system may be expressed in various forms, viz. 
the quotient of any two of the expressions x +p, y + q, z + r, or of any two of the