838 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
we regard the elements of the elliptic orbit as constant, the X-Y-Z functions are 
expressible as a number of simple time-harmonics. But in § 4, where the state of 
tidal distortion due to Diana was found, they were assumed to be so expressible ; 
therefore that assumption was justifiable, and the remainder of that section concerning 
the formation of the disturbing function is applicable. 
The problem may now be simplified by the following considerations :—The equation 
(12) for the rate of variation of the ellipticity of the orbit involves only differentials of 
the disturbing function with regard to epoch and perigee. It is obvious that in the 
disturbing function the epoch and perigee will only occur in the argument of 
trigonometrical functions, therefore after the required differentiations they only occur 
in the like forms. Now the epoch never occurs except in conjunction with the mean 
longitude, and the longitude of the perigee increases uniformly with the time (or 
nearly so), either from the action of other disturbing bodies or from the disturbing 
action of the permanent oblateness of the planet, which causes a progression of the 
apses. Hence it follows that the only way in which these differentials of the 
disturbing function can be non-periodic is when the tide-raiser Diana is identical with 
the moon. Whence we conclude that— 
The tides raised l>y any one satellite can produce no secular change in the, eccentricity 
of the orbit of any other satellite. 
The problem is thus simplified by the consideration that Diana and the moon 
need only be regarded as distinct as far as regards epoch and perigee, and that they 
are ultimately to be made identical. 
Before carrying out the procedure above sketched, it will be well to consider what 
sort of approximations are to be made, for the subsequent labour will be thus largelv 
abridged. 
From the preceding sketch it is clear that all the terms of the X-Y-Z functions 
corresponding with Diana’s tide-generating potential are of the form 
(« fi- 5e -{- ce 2 +cZe 3 -f-jfe 4 -fi&c.) cos [Zy-fi mlflt -j-e)fi- nzs +S]. 
From this it follows that all the terms of the functions are of the form 
F(«+ be +ce 2 +cZe 3 +/'e 4 +&c.) cos [Zy+ m(flt +e)-j- n-cs -f-S —f~\. 
Also by symmetry all the terms of the X -Y -Z' functions are of the form 
(a+be -f ce 2 -f cZe 3 +/e 4 +&c.) cos [Zy 7 + m(flt -j- e') + nzs' ff- S], 
and in the present problem the accent to y may be omitted. 
The products of the jl’-I) it- functions multiplied by the X -Y -Z' functions occur in 
such a way that when they are added together in the required manner (as for example 
in Y'Z' 332- + X / Z / ^2-) only differences of arguments occur, and y disappears from the 
disturbing function. Also secular changes can only arise in the satellite’s eccentricity 
and mean distance from such terms in the disturbing function as are independent of 
and ct, when we put e' = e and A = Hence we need only select from the 
