DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
93 
and if the values of x-, y\ given by (I.) be substituted in this expression, it becomes 
„'~ + [Z ,«] (32.) 
Let u — [p, q], then (making use of (31.)) 
[p> ?]-[!’ «]+[* f] + [ z ’ 
Now suppose that, by virtue of the differential equations (I.), the values of p and q 
are constant ; or, in other words, that 
p=p(x 1} &c., y x , &c., t) 
q — ^(x „ &c., 3 /„ &c., 0 
are any two integrals whatever of the system (I.); p, q representing two arbitrary 
constants. The equation p '= 0 gives (see (32.)) 
*+[Z,p]=0. 
or 
hence 
In like manner 
3’ ?]=-[[ z ’ ?]=[?> [ z - ri]- 
l>’ §] =fr> b. z l]- 
Thus the expression given above for \_p, q~]' becomes 
?]' -[p, £?, z ]]+[?> [ z > p]]+[ z > [p> ?]]» 
which is identically equal to 0, by the theorem (30.). Consequently, for any two 
integrals p and q, 
\_p, q] = constant (33.) 
This theorem, as has been already mentioned, was discovered, in the case of the 
dynamical equations, by Poisson ; and the fact that he was able to arrive at it 
through so long and complex a process as that which he has given in his first memoir 
on the Variation of Arbitrary Constants*, must be looked upon as a remarkable 
instance of his analytical skill. I am not acquainted with any attempt to simplify 
the demonstration, except that of Sir W. Hamilton^; in fact it is probable that no 
material simplification was attainable without the help of the transformation of the 
differential equations to the form (I.), towards which Poisson (as Jacobi has remarked) 
only made a first step. Sir W. Hamilton’s demonstration may certainly be con- 
sidered simple as compared with that of Poisson. That which I have given above 
will, I hope, be regarded as a further improvement. 
23. In what follows I shall use such expressions as “the integral c,” as an abbre- 
viation for “the equation c = <p(x„ &c., y x , &c., t).” 
* Journ. de l’Ecole Polytechnique, tom. viii. f Philosophical Transactions, 1835, p. 108-9. 
