ANALYSIS TO THE DYNAMICAL THEORY OF THE TIDES. 
255 
Suppose the system starts from rest in its position of relative equilibrium, so that 
^ = 0 when t = 0, or ^ = 0. Then 
TJj -j- 2 cl>^Vq — 0 
Vj — 2wp,Uo = 0, 
whence, 
Ui 
and therefore 
^o=-2i = » 
TT = Yj_ _ _ 
^ 2(oii Aora d/x 
The last equation gives 
(1 - h 
fjr dfi 
4ft)V 
a 
H 
% 
/i — 
0 
a 
no arbitrary constant being added, since both sides vanish when p = dh !• Thus, 
, 0 ?, 2M „ 
fl r;— = - aixr, 
dfi 
or 
^ 2o)~a~a. r , 
Suppose for example that h is constant; we shall then obtain 
(o^a-'ct 
Cl = ^ + const 
— 1 
fjh 
0 o 
(o^a'^oc 
{'h + + + const. 
Choosing the constant so that the mean value of Ci over the surface is zero, we 
obtain finally 
o o 
(o-a~x . 
/ — 1 ^ ^ <• 8_p I 4 P ^ 
gh i3 5-t^4+ 
Hence the particular solutions of the ditferential equations which represent the 
“ forced ” motion due to the disturbing influence in question are 
