DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
95 
terms of c 1} &c. are not all independent; a supposition which we exclude, in order 
that k i} ... k m may represent m distinct integrals. 
The equations above written lead obviously to the following conclusions: — 
(1.) If f be a given function of the m constants c,, ... c m ; then the determination 
of another function g, such that [/, g] =0, depends in general upon the solution of 
a linear partial differential equation of the first order. 
(2.) It is impossible that the conditions [Zr t -, Ay]=0 can exist for every binary com- 
bination of k 1} ..., k m , unless [c i3 cj=0 for every binary combination of c„ ..., c m . 
25. As an illustration of the first of these conclusions, we may take a case which 
actually occurs in many dynamical problems. Let c„ c 2 , c 3 be three integrals, such 
that 
[c„ c 3 ^|— c 19 ^c 3 , cj — c 2 , [c 13 c 2 J— c 33 (c.) 
and let it be required to find a function g of c 13 c 23 c 33 such that [c 13 p]=0. The 
equation (L.) of the last article gives, if we put f—c x , and introduce the conditions (c.). 
The solution of which is 
c ^L-r^L- 0 
C 3 j L 2 
ClCa ClC o 
g=^{c\+cf), . . (g.) 
being an arbitrary function (which may evidently also contain c, in an arbitrary 
manner). 
If, instead of f—c x , we put it will be found that the expression on 
the right of the equation (L.) vanishes identically; so that in this case, if g be any 
arbitrary function of c 1} c 2 , c 3 , the condition \_f, g~] =0 will be satisfied. 
26. If a„ « 2 , ... a n , b l} b 2 , ... b n be a system of normal elements (art. 20.), we have 
(equation (25.), art. 9.) 
[/,?] 
— v Af,g) } 
1 difli, bi )’ 
where f, g represent any two functions of the elements, or in other words, any two 
integrals whatever. If in the above equation we put successively f=a i ,f=b i , we 
obtain 
dcii 
(36.) 
In the case where the principle of vis viva subsists, we may suppose the constant of 
vis viva, h, to be one of the elements. In this case (see (29.), art. 19.) the element 
conjugate to h is r, and t appears in none of the integrals explicitly, except one, 
namely, the integral conjugate to h, which is 
If, then, g be any integral whatever, not containing t explicitly, it cannot contain t, 
since any combination of the normal integrals involving r, will involve it in the form 
