ON THEORETICAL DYNAMICS. 29 
moir shows, these cases are far more numerous than those to which the 
method applies; thus for example the equation of vis viva, combined with 
any other integral whatever, leads to an illusory result. The theorem of 
Poisscn may lead to an illusory result in two ways; either the resulting 
integral may be an identity O=0, or it may be an integral contained in the 
integrals already known, and which consequently does not help the solu- 
tion. It appears by the memoir that the two cases are substantially the 
same, and that it is sufficient to study the case in which the two integrals 
lead to the identity O=0. Suppose that one integral is given, the author 
shows that there always exist integrals which, combined with the given 
integral, lead to an illusory result, and he shows how the integrals which, 
combined with the given integral, lead to such illusory result, are to be 
obtained. For instance, in the case of a body moving round a fixed centre, 
there are here two known integrals ; first, the equation of vis viva (but this, 
as already remarked, combined with any other integral whatever, leads to an 
illusory result); secondly, the equation of areas. 
The question arises, what are the integrals which, combined with the 
equation of areas, lead to an illusory result? The integrals in question 
are, in fact, the other two integrals of the problem; so that the inquiry into 
the integrals which give an illusory result, leads here to the completion of 
the solution. The like happens in two other cases which are considered, 
viz. 1. the problem of two fixed centres, and the problem of motion 7m plano 
when the forces are homogeneous functions of the coordinates of the degree. 
2. Indeed the case is the same for all problems whatever, where the co- 
ordinates of the points of the system can be expressed by means of two inde- 
pendent variables. 
The next problem considered is that where two bodies attract each other, 
and are attracted to a fixed centre. Suppose, first, the motion is iz plano, 
then as in the former case al/ the integrals will be found by seeking for 
the integrals which, combined with the equation of areas, give an illusory 
result. When the motion is in spate, the principle of areas furnishes three 
integrals (the equation of vis viva is contained in these three equations) ; 
the integrals which, combined with the integrals in question, give illusory 
results, are eight in number, and, to complete the solution, there must be 
added to these one other integral, which alone does not put the method in 
default. The problem of three bodies is then shown to be reducible to the 
_ last-mentioned problem ; and the same consequences therefore hold good 
with respect to the problem of three bodies, viz., there are eight integrals 
which, combined with the integrals furnished by the principle of areas, give 
illusory results. To complete the solution it would be necessary to add to 
these a ninth integral, which alone would not put the method into default. 
57. The author remarks that it appears by the preceding enumeration 
that the method of integration, based on the theorem of Poisson, is far 
from having all the importance attributed to it by Jacobi. The cases of 
exception are numerous; they constitute, in certain cases, the complete 
solutions of the problems, and embrace in other cases eleven integrals out 
of twelve. But it would be a misapprehension of his meaning to suppose 
that, according to him, the cases in which Poisson’s theorem is usefully 
applicable ought to be considered as exceptions. The expression would not 
be correct even for the problems, which are completely resolved in seeking 
for the integrals which put the method into default; there exists for these 
problems, it is true, a system of integrals which give illusory results; but 
these integrals, combined in a suitable manner, might furnish others to 
which the theorem could be usefully applied. 
