655] 
A MEMOIR ON DIFFERENTIAL EQUATIONS. 
115 
50. Supposing that any two integrals are c', d', so that a general integral is 
cf) (H, a, b, c!, d'), then c', d' qua functions of H, a, b, c, d, e are integrals of the 
former equations (H, 6) = 0, (a, 6) = 0, so that again changing the notation, and writing 
c, d instead of the accented letters, we have (H, a, b, c, d) as solutions of the three 
equations (H, 9) = 0, {a, 6) = 0, (b, 6) = 0, viz. a being any solution of the first equation, 
and b any solution of the first and second equations, we see how to find two others 
c, d, of the same two equations, which are such that 
(H, a) = 0, (H, b) = 0, (H, c) = 0, (H, d) = 0, 
(a , b) = 0, (a , c) = 0, (a , d) = 0, 
(b , c) = 0, (b , d) = 0; 
or, attending only to the integrals H, a, b, c, these are integrals of the equations 
(.H, 9) — 0, {a, 9) = 0, ([b, 9) = 0, such that 
(H, a) = 0, (H, b) = 0, (H, c) = 0, {a, b) = 0, {a, c) = 0, (b, c) = 0. 
We here say that H, a, b, c are a system of conjugate solutions. Attempting to 
continue the process, it would appear that there is not any new independent integral d, 
such that (.H, d) = 0, {a, d) = 0, (b, d) = 0, (c, d) = 0 (the first three of these are 
satisfied by the integral d found above, but the last of them is not); we may, 
however, taking d an arbitrary function of H, a, b, c, replace H by d\ viz. we thus 
have the four integrals a, b, c, d, such that 
{a, b) = 0, (a, c) = 0, (a, d) = 0, (b, c) = 0, (b, d) = 0, (c, d) = 0, 
and which are consequently said to form a conjugate system. 
51. The process is of course general, and it shows how, in the case of a 
Hamiltonian system of 2n variables, it is possible to find a system H, a, b,..., f con 
sisting of H and n — 1 other integrals, or, if we please, a system of n integrals 
a, b,..., f g, such that the derivative of any two integrals whatever of the system is 
= 0; any such system is termed a conjugate system. 
Hamiltonian System—the function V. Art. Nos. 52 to 58. 
52. Taking a Hamiltonian system with the original variables x, y, z, p, q, r, we 
adjoin the two new variables t, V, forming the extended system 
dx 
dy 
dz 
dp 
dq 
dr dV 
dH~ 
dH 
~dH 
dH~ 
dH 
dà dt ~ dH dH dH 
dp 
dq 
dr 
dx 
dy 
dz P dp ^ ^ dq ^ 1 dr 
Supposing the integrals of the original system to be a, b, c, d, e, we have 
H = H (a, b, c, d, e) a determinate function of these integrals ; also an integral 
T = t—(f)(x, y, z, p, q, r) and an integral \= V—-yJr(x, y, z, p, q, r); these integrals, 
exclusive of the last of them, serve to express x, y, z, p, q, r as functions of 
a, b, c, d, e, t — t ; and the last integral then gives V = X. + a function of the last- 
mentioned quantities. 
15—2