94 
PROFESSOR DONKIN ON THE 
It is of course understood that the function on the right contains neither c nor any 
other arbitrary constant explicitly. 
Let then f, g be any two given integrals of the system (I.). It has been shown 
that we shall always have 
[/, g] = constant (K.) 
But this equation may be true either (1) identically, or (2) not identically. In the 
hrst case the expression \_f, g] may either be identically =0, or it may reduce itself 
identically to a determinate constant , which might always be made unity by multiply- 
ing one of the integrals by a factor. (In the case of a “normal system ” of integrals 
(art. 20.), it has been seen that every binary combination gives either 0 or 1.) But 
if the above equation (K.) be not identically true, so that \_f g] obtains a constant 
value only by virtue of the differential equations, then the constant on the right of 
(K.) is an arbitrary constant , and that equation is itself an integral. But here again 
there are two cases ; for the function \_f, g] may be only a combination of the func- 
tions on the right of the two integrals / 5 g ; and then (K.) is not a new integral, but 
only a combination of the two given ones ; or, on the other hand, [/, g] may be a 
function independent of /, g ; and then (K.) is really a new integral, which cannot 
be produced by merely combining the other two. Thus it appears that Poisson’s 
theorem may in some cases lead to the discovery of new integrals, when two are known. 
On this subject, and others connected with it, I refer to the interesting memoir of 
M. Bertrand in Liouville’s Journal (1852), “Sur Integration des equations differ- 
entielles de la Mecanique.” 
24. Let c„ c 2 , ... c m be any m integrals, and let f, g be any two functions of the m 
constants c 1} c 2 , ..., c m , so that f g are also two integrals; and considering f, g as 
functions of c„ ..., c m , and, through them, of the variables, we have exactly as in 
art. 9, equation (24.), 
the summation extending to all binary combinations of the m constants c„ &c. If 
then we suppose k u k 2 , ..., h m to be m functions (such as h, k) of the m constants 
c 13 ..., c m , we shall have for any pair k p , k q , 
(34.) 
(the summation referring as before to i,j ) ; and the inverse equations (obtained either 
by considering c„ &c. as functions of k x , & c., and reasoning in the same way, or by 
multiplying the above equation by ^ and summing with respect to p, q) will be 
= ( 35 '> 
(the summation referring to p, q). 
This inversion can only fail in the case in which the equations expressing k x , & c. in 
