158 REPORT—1899. 
therefore only consider this latter class of integrals. It is shown that 
integrals of this class can be compounded of another kind of integrals, 
called by Bruns homogeneous. When a homogeneous integral is resolved 
into factors which are rational integral functions of the quantities y, it 
appears that each of these factors either is itself an integral or can be 
made into an integral by associating certain factors with it; and so, 
finally, every integral of the problem of m bodies which is algebraic in the 
variables x, y, and is independent of ¢, can be compounded algebraically 
from integrals of a very special class, which are rational integral functions 
of the quantities y, and rational functions of the quantities x and c, and 
which are, moreover, homogeneous. It is further shown that if ¢ denote 
an integral of this last class, and ¢, denote the terms in it which are of 
highest order in the quantities y, then o involves the «’s rationally and 
integrally, and only by means of the expressions (y,%,—y, “,), and 5 
does not involve c. 
Now let A, B, C be the three components of angular momentum of 
the system, let L’, M’, N’ be the three components of linear momentum, 
let L, M, N be the coordinates of the centre of gravity, and let 
A’=MN’—NM’, B’=LN’—-L’'N, C’=LMW’—LM ; 
all these quantities are supposed to be expressed in terms of the quanti- 
ties x and y,so that any one of them equated to a constant represents 
one of the known integrals of the system of differential equations. Then 
it is shown that ¢) involves the variables x only in the combinations 
A, B,C, A’, B’, C’, and is a rational integral function of A, B, C, A’, B’, C’, 
and the y’s ; and then ¢, is proved to be a rational integral function of 
fasts, OC, A’, BGC’, L’, WY, N., T, aay. 
y= (A; B, C, AG Bi, C5 L, MW’, N’, T). 
Now let —U be the potential energy of the system, so T—U is an 
integral. Then the quantity 
J=AA, B, 0, A’, BY, C’, L’, M’, N’, TU) 
is also an integral, since it is compounded solely of integrals ; and when 
it is arranged according to powers of the variables y it coincides with the 
integral ¢ in the terms of highest degree. The difference 
y=o—T 
is therefore an integral of the same kind as ¢, except that its degree in 
the variables y is at least one unit lower than the degree of ¢ in these 
variables. Thus any integral @ can be made to depend on the known 
integrals and an integral of lower degree in the y’s: proceeding in this 
way, @ can be made to depend on the known integrals and an integral of 
the same kind as ¢, but of zero degree in the y’s ; but such an integral 
would be a constant. Thus Bruns arrives at the theorem : In the problem 
of n bodies, the only integrals which involve the coordinates and velocities 
algebraically, and which do not involve the time explicitly, are compounded 
of the integrals of the centre of gravity, of angular momentum, and of 
energy. 
Bruns then proceeds to the reduction of the differential equations of 
the problem of three bodies which has already been given in § 1 of this 
report, and shows that the system of the 6th order at which he arrives has 
