130 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
[655 
whence 
dV — pdx - qdy — r dz = d\+ (t — r) dH + a'da + b'db + c'dc — a da' — bdb' — c dc ; 
or, in place of a, b, c, a', b\ o', introducing a, /3, y, 8, e, and attending to the value 
of H, 
dV — pdx— qdy— rdz= d\+ (t — t) dH + l dH + ^^d8+ ^-de. 
77. Suppose H, 8, e absolute constants, this becomes 
d ( V — X) = p dx + q dy + r dz, 
or 
F= A, + J (pdx + qdy + r dz), 
and we have thus a solution of the partial differential equation 
p 2 + q 2 + r 2 = x 2 + y 2 + z 2 4- 2 H; 
viz. p, q, r are here to be determined as functions of x, y, z by the equations 
p 2 + q 2 + r 2 = x 2 + y 2 + z 2 + 2 H, 
y +q = 8(x+p), 
z +r = e (x +p). 
We have 
iIH + x 2 + y 2 + z 2 = p 2 + {y — 8 (x + p)} 2 + {z — e (x+p)} 2 ; 
or, on the right-hand side, writing p 2 = (x + p) 2 — 2x(x+p) + x 2 , 
„ left „ „ x 2 = (x—p) 2 — 2x (x +p) +p 2 , 
the equation is 
(1 + 8 2 + e 2 ) (x + p) 2 — 2(x+8y + ez) (x + p) — 2H — 0, 
which gives |) as a function of x, y, z. But the result is a complicated one, except 
in the case H = 0; we then have 
2(x+8y + ez) 
X+ P~ l + ^+e 2 ’ 
2 8(x+8y + ez) 
y ^ q l + S 2 + 6 2 ’ 
z + r = 
2e (# + Sy + ez) 
1 + 8 2 + e 2 
and thence 
V=\-%(x 2 + y 2 + z 2 ) + (0C ~~+ y 2 + € * ) , 
a complete solution of the partial differential equation 
p 2 + q 2 + r 2 = x 2 + y 2 + z 2 .