574 ME. G. H. DARWIN ON PROBLEMS CONNECTED 
The surface value of p when r—a is 
wvcP 79 Y . t N 
(55) 
where p 0 is given by (34). 
If we write —^n—e for e we see that a term Y sin (yt —e) in the effective disturbing 
potential will give us 
.( 56 ) 
wva° 
— 79 Y 
P Po y 2 2 3 19 3 r 2 C ° S 6 
Now suppose wr ~S cos vt to be an external disturbing potential per unit volume of 
the earth, not including the effective potential due to gravitation, and let r=a+cr / 
be the first approximation to the form of the tidal spheroid. Then by the theory of 
tides as previously developed (see equation (15), Section 5, “ Tides”) 
°7 S / . \ i , 19ar 
= - cos e cos (vt — e), where tan e= -- 
a g x ' Agaio 
Then when the sphere is deemed free of gravitation the effective disturbing 
potential is wr~i S cos vt— g—) ; this is equal to — wr 3 sin e S sin (vt — e ). 
Then in proceeding to a second approximation we must put in equation (56) 
Y = — wr" sin e S. 
Thus we get from (56), at the surface where r—a, 
, w%m * 19 . c , N 
P=Po~\—~T ' sm 6 b cos K vt — e ) • • 
• (57) 
To find p 0 we must put r=« + cr as the equation to the second approximation. 
Then p 0 is the surface radial velocity due directly to the external disturbing potential 
ivr "\S cos vt and to the effective gravitation potential. The sum of these two gives an 
effective potential ivr z (S cos which is the Y' cos vt of (34). 
Then p 0 is found by writing this expression in place of Y' cos vt in equation (34), and 
we have 
5 wcfii ~ <r\ 
P°=MU( Scoavt -$} 
Substituting in (57) we have 
bwcP 
biovcP 79 
U WU' / tV Vtl/ 4 V • / \ \ 
P= b cos vt - 5,7+ YYT TYhi S1U e b cos \ vt ~ e )) • 
19 
V A.O.0- 
(58) 
