[655 
655] A MEMOIR ON DIFFERENTIAL EQUATIONS. 127 
stems H, a, b; 
iese, considering 
at to the first- 
where the term in d6 is 
= ddd(-J + . 1 _ + , 1 ), 
\0 + CL 6 + /3 6 + y ) 
which, from the equation which determines 6, is 
>w consider the 
it and p, q, r 
solution is that 
¡rmined by the 
~ (dx dy dz\ 
= 0—+ -* + —, 
\x y z) 
and the value of dV is thus 
^t±Z dx+ ?±i dy+ e _±l di , 
X y J z 
r symmetry the 
The solution contains apparently the four constants A, a, /3, y, but there is no loss of 
generality in writing, for instance, a = 0, and the number of constants really contained 
in the solution may be regarded as 3. 
This is there- 
as such, is a 
73. To show how the equations H = const., a=a 0 , b = b 0 , c = c 0 , d = d 0 , e = e 0 
give a solution; remarking that these equations are pqr — 1 = H, p=p 0 , q = q 0 , r=r Q , 
qy — px = q 0 y 0 — p 0 x 0 , rz-px = r 0 z 0 -p 0 x 0 , we find 
p(x-x 0 )=q(y- y 0 ) = r (z - z 0 ), 
and consequently p, q, r = 
ery elegant one 
- a in place of 
^/(i + H)( y ~ 2/0)3(z ~ Zo) ~, + Zo)H 7 Xo)s , j/(i+H)^ (y 7 2/0)3 , 
0 - O 3 (y - yoY (z - z 0 y 
respectively: whence 
V = \ + j(pdx + qdy + r dz), 
= \ + 3j/(l+H)(x-x 0 f {y - y 0 f (z - z$, 
which is the solution involving the four constants A, x 0 , y 0 , z 0 . 
function of the 
If in the foregoing value of V we consider x 0 , y 0 , z 0 as variables, then p, q, r 
having the values just mentioned, and p 0 , q 0> r 0 being equal to these respectively, we 
obviously have 
dV = p dx + q dy + r dz — p 0 dx 0 — q 0 dy 0 — 7' 0 dz 0 . 
lg 6 as a neiv 
e of dV is 
74. Considering now the augmented Hamiltonian system, we join to the foregoing 
integrals a, b, c, d, e, the new integrals t — t = ^ and V — \ — 3px. And then expressing 
all the quantities in terms of t — t, 
x = bc(t — t), 
y =ca(t -T) + p 
z = ab(t — t) + - . 
v c 
p = a, q = b, r = c, H = abc — 1, 
V = A + 3abc (t — t). 
e of dV is