PROFESSOR CHAELIS ON THE PROBLEM OF THREE BODIES. 
541 
(4) The expressions for E and EE show that these quantities contain terms which 
have t for a factor, and may therefore increase indefinitely. This circumstance creates 
no difficulty with regard to IT, which is always part of a circular arc affected by a 
sine or cosine. But as E appears as a coefficient, it might seem that the develop- 
ments of r and 6 contain terms which admit of indefinite increase. It must, how- 
ever, be observed, that according to the remark made at the end of art. 12, the 
function that has given rise to these terms is really affected by a cosine, and that 
they have their origin in the development of that function in terms arranged accord- 
ing to the disturbing force. 
The following considerations will enable us to obtain, at least approximately, the 
periodic functions of which 11 and E are partial developments. Whatever functions 
the complete values of IT and E are of t, they may be expanded in series of the form 
the two first terms of which are already determined. Hence 
£?n „ , „ , , 0 
:^'_|-2y74-&C. 
Let t be indefinitely small. Then substituting 
1 
2n da^ 
B for-- 
na da 
dn 
7E 
we shall have strictly the values of and for the epoch at which t commences 
dt 
VIZ. 
m 
dt 
=B 
De' 
COS(to'- — raj 
De' 
— 2 sin — '^')- 
Anea‘^ ' ' 
dU 
Now if t commenced at a different epoch, we should obtain for and ^ the same 
expressions as those above, but different in value, because by hypothesis these 
differential coeffieients vary with the time. The changes of value, which in actual 
cases take place very slowly, are due to changes in the eccentricities, and in the lon- 
gitudes of the apses, of the two orbits, and will be very approximately taken into 
account by substituting in the above equations for e, e', ra and ra-', the variable quan- 
dYV 7E' 
titles E, E', n and 11'. • Like considerations apply to the values of -yr and — . Thus 
JiX\^ llo I VJ Ci t I/IJV/ VCtlUCo Cillvi 
we shall have four differential equations, by the simultaneous integration of which the 
four quantities maybe obtained as periodic functions of the time. The arbitrary 
constants introduced by the integration are determined by the known values of the 
functions when ^=0. These periodic functions are to be substituted for E, E', 11 
and II', wherever these quantities occur in the developments of r, r', 6 and & . The 
four equations just mentioned are identical with those obtained by the method of the 
variation of parameters for determining the eccentricities and longitudes of the 
apses. It is worthy of remark, that in both methods the changes of the eccentricities 
and of the longitudes of the apses which are due to the disturbances, are taken into 
account in calculating the changes themselves, so that the approximation does in 
4 B 2 
