458 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Again for the second addition when n— 0, we have 
tp¥= 3 <f) 
P¥ (p°- - (f) + \pq (jr - if) (I - 6/a/) = \pq (jp - f ) 3 
W+IpV= ip? 5 ( 3 p 3 +r) 
fp 5 ? 3 + |//v/ = Jj/v/, 
So that 
G-P — = — \E x p l q s ' n 2 e l -\-Ep ?, (f(p i —(f) sin 2e-\-^E. 2 pq I sin 2e 3 
—\E\ P 5 ?(p 2 +3? 3 ) sine^-j -\E’pq{p l —(ff sin P + \E' 2 pcf (Spr<f) sin eh 
+§ E"p s q B sin 2e".(16) 
And 
It 
C 
= F sin n + G cos n 
(17) 
To find M it is only necessary to substitute n —- for n, and we have 
u 
m 
c 
= — F cos G sin n 
(18) 
Now there is a certain approximation which gives very nearly correct results and 
which simplifies these expressions very much. It has already been remarked that the 
three <fi-tides have periods of nearly a half-day and the three ^P-tides of nearly a day, 
and this will continue to be true so long as fl is small compared with n; hence it may 
be assumed with but slight error that the semi-diurnal tides are all retarded by the 
same amount and that their heights are proportional to the corresponding terms in the 
tide-generating potential. That is, we may put e 1 =e 3 =e and Ep=-E 2 —E. The 
similar argument with respect to the diurnal tides permits us to put e' l = e' 2 =e / and 
E\=E',z=E'. 
Then introducing the quantities P—p^—cp— cos i, Q=2pq= sin i and observing 
that 
ip 5 q(p~ - 3 ? 3 )+¥?(¥ - f)(p 4 +?'~ G pY) + hiYAf - r) = p ?( p 3 - <f) ( 1 - ¥¥) 
=\PQA-m 
iYqip'+Y) — ip?(p 3 —? 3 ) 3 —ip? 5 (3p 3 +? 3 ) =ip?(p 3 —? 3 )(1 + 2y>V— 1 + 4pV) = sPQ 5 
we have, 
