A MEMOIR ON DIFFERENTIAL EQUATIONS. 
109 
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Supposing that the set is (x, y, z, p, q, r), so that there are in all five integrals 
a, b, c, cl, e, then the theorem may be otherwise stated, we have (a, b) a function of 
the integrals a, b, c, d, e. 
Observe that, knowing only the integrals a and b, we find (a, b) as a function of 
x, y, z, p, q, r, this may be = 0, or a determinate constant, or it may be such a 
function that by virtue of the given values of a and b it reduces itself to a function 
of a and b; in any of these cases the theorem does not determine a new integral. But 
if contrariwise the value of (a, b), obtained as above as a function of the variables, is 
not a function of a, b, then it is a new integral which may be called c. 
38. To obtain in this way a new integral, we require two integrals a, b other 
than H; for knowing only the integrals a, II, the theorem gives only {a, IT) an 
integral, and we have of course (a, H) = 0, viz. we do not obtain a new integral. 
But starting from two integrals a, b other than H, we may obtain as above a 
new integral c; and then again {a, c) and (b, c) will be integrals, one or both of 
which may be new. And it may therefore happen that in this way we obtain all the 
independent integrals a, b, c, d, e; or the process may on the other hand terminate, 
without giving all the independent integrals. 
The theory is obviously applicable throughout to the case of a bipartite set 
(x, y, z,..., p, q, r, ...) of 2n variables. 
39. It may be remarked here that, in the Hamiltonian system, a value of the 
multiplier is M = 1 ; and consequently, if in any way all but one of the integrals, 
that is, 2n — 2 integrals, be known, the remaining integral can be found by a 
quadrature. 
It is further to be noticed that, if we adjoin a new variable t and a term = dt 
to the system of equations; then the 2n — 1 integrals of the original system being 
known, all the original variables can be expressed in terms of the 2n — 1 integrals 
regarded as constants and of one of the variables say x: we then have 
or 
or say 
viz. if after the integration we suppose the 2n — 1 integrals replaced each of them 
by its value, we have 
e=t-(f>(x, y, z, ..., p, q, r, ...), 
which is the remaining or 2wth integral of the original system as augmented by 
the term = dt.