[655 
quations imply 
), 
instead of p, x. 
plies the other, 
are 
cIk da B 
dxdp’^’ 
Conjugate Integrals of the Hamiltonian System. Art. Nos. 46 to 51. 
46. For greater clearness, let n = 4, or let the variables be x, y, z, w, p, q, r, s; 
the system of differential equations therefore is 
dx 
dy 
dz 
dw 
dp 
dq 
dr 
ds 
dH 
dH 
dH~ 
dH " 
dH~ 
dH 
dH 
dH’ 
dp 
dq 
dr 
ds 
dx 
dy 
dz 
dw 
and any integral hereof is as before a solution of (H, 6) = 0. Assume that the 
integrals are H, a, b, c, d, e, f so that 
(H,a) = 0, (H,b) = 0, (£T, c) = 0, (H, d) = 0, (H, e) = 0, (H,f) = 0. 
Considering here a as denoting any integral whatever, that is, any solution 
whatever of the partial differential equation (H, 0) = 0, it is to be shown that it is 
possible to determine 6 so as to satisfy as well this equation (H, 6) — 0, as also 
the new equation (a, 6) = 0. 
47. We, in fact, satisfy the first equation by taking 
0, =6 (H, a, b, c, d, e, f), 
any function whatever of the seven integrals. But, 6 having this value, we find 
/ m / tt\ d0 , .dd , lx dd , . dd . n .d6 , .d6 . d6 
(a, 6) = (a, + *>)&+<* ')* + («■ d >3S + (“' ^de+^^df' 
15 
c. x.