PROFESSOR CHALLIS ON THE PROBLEM OF THREE BODIES, 
527 
The plane of xy has hitherto been supposed to be coincident with the plane of m’s 
orbit at the time To. On this supposition values of r and & have been obtained, which 
fully take into account the first power of the disturbing force and the mutual incli- 
nation of the orbits. We are now at liberty to suppose the plane of xy to have any 
other position making a small angle with the planes of the orbits of m and m! in their 
positions at the epochs To and TJ,, and the equation (3.), viz. 
d'^z d'R 
dt‘^'' r^'dz ’ 
may be employed for finding a series for z in terms of t. In the second term of this 
equation the value of r given by the second approximation is to be substituted, but 
in the third term it is only required to substitute the values of r, 9, r' and 9’ given by 
the first approximation. 
Also for z and z' we may substitute in the functions of 
t which express the values of these quantities on the supposition that the motions are 
undisturbed, and that they are referred to the new plane of xy. Thus becomes 
a function of t and constants, and the above equation takes the form 
d^z 
which admits of being integrated only by successive approximations. 
The process by which it has been shown that r, 9 and z are approximated to, gives 
at the same time, mutatis mutandis, the values of r' , 9' and z' to the first power of the 
disturbing force of m. By means of these six quantities the approximation may be 
carried to terms inclusive of the square of the disturbing forces. 
Having thus exhibited the general scheme of this approximate solution of the 
Problem of Thi’ee Bodies, I proceed to exemplify its practicability. 
First Approximation. 
4. The first approximation, which omits the terms involving the disturbing force, 
and is therefore identical in form with the solution of the problem of two bodies, is 
obtained by integrating the equations 
dt 
The first equation gives, by integration. 
w(^-l-T) = cos ^x/aV— (a— r)S 
where for the sake of brevity a is put for for 1 — and n for ^or^^. Let g 
it- aM 
be the constant introduced by the integration of the second equation, and in order 
to designate the constants in the present problem by the letters usually employed in 
