214 " REPORT—1862. 
= ) (ante, No. 87); § V. to the problem of two centres (ant2, No.62), and § VI. 
to the problem of three bodies (post, No. 117). 
111. Donkin’s memoir “ On a Class of Differential Equations &c.” (1855). 
Part I. Nos. 27 to 30 relate to the problem of central forces (in space), No. 31 
to the rotation of a solid body, and $ III. to the same subject, viz. Nos. 40 
and 41 to the general case, Nos. 42 to 44 to the particular case A=B; 
and Nos. 45 to 48 to the reduction thereto of the general case by treating 
the forces which arise from the inequality of A and B as disturbing forces. 
Part II. Nos. 59 and 60 relate to the spherical pendulum ; Nos. 72 and 73 to 
“ Transformation from fixed to moving axes of coordinates,” say to Relative 
Motion ; and Nos. 84 to 96 to the problem of three bodies ( post, No, 120). 
The Problem of Three Bodies, Article Nos. 112 to 123. 
112, A system of differential equations, such as 
dx,_dv, dn 44 
x, 2 Xn41 
(n equations between n+1 variables), may be termed a system of the nth 
order, or more simply a system of n equations. Let (u,, u,...-U,4,) be 
any functions of the original variables (w,, v,,....w,4,), the system may be 
transformed into the similar system 
du,__du, du, 
U, oes SP Th 
and if it happens that we have e.g. U, identically equal to zero, then the 
system becomes 
0 (ee _M, du, +) 
off.) 
n+l 
so that we have an integral u,=c, and then in the remaining equations 
substituting this value, or treating wu, as constant, the system is reduced to 
one of (m—1) equations. Or again, if it happen that we haye in the trans- 
formed system m equations (m<n), say 
‘ _ Mh +1, 
Uy Be Vines 
which are such that U,, U,...U,,,, are functions of only the m+1 variables 
Uz, Uz+++Un4y» then the integration of the proposed system of n equations 
depends on the integration in the first instance of a system of m equations. 
It is to be observed that if the system of m equations can be integrated, 
then the completion of the integration of the original system depends on the 
integration of a system of »—m equations, and in this sense the original 
system of n equations may be said to be broken up into two systems of m 
equations and n—m equations respectively : but non constat that the system 
of m equations admits of integration ; and it is therefore more correct to say 
that, from the original system of the n equations, there has been separated off 
a system of m equations. 
113. The bearing of the foregoing remarks on the problem of three bodies 
will presently appear. It will be seen that whereas the problem as it stood 
before Jacobi depends on a system of seven equations, it has been shown by 
him that there may be separated off from this a system of sia equations. 
