514 MB, G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
generating potential of the third order due to the moon will raise tides in the earth, 
when there is a frictional resistance to the internal motion, given by 
7 TCI 
a IOqc 
— " 12 ^ sin 3 $ sin 3(<£—w+/) -j -\F' sin 9 (1 — 5 cos- 9) sin (<£ —w- \-f) 
Now <x is a surface harmonic of the third order, and therefore the potential of this 
layer of matter, at an external point whose coordinates are r, 9, <£, is 
4 
- 7TCIVJ 
[ay 3 Mo? 
\rj <J ~~7 ?’ 4 a 
distorted spheroid exercises on a particle of mass to, situated at r, 9, <f>, is 
Hence the moment about the earth’s axis of the forces which the attraction of the 
3 Mrna da 
v r 4 <7</>' 
Now if this mass be equal to that of the moon, and r—c, then | 1 A — h -Me ?—- - C, 
^ 7 r 4 7 e 7c’ 
where, as before, C is the moment of inertia of the earth. 
Hence the couple JdU, which the moon’s attraction exercises on the earth, is given 
by —E-Cyy, where after differentiation we put 9=^ and <£=^4" w - 
Now 
—' °-=y 6 ^ a ~- [-fi^siiffdcos 3(<£ — w+/)— ^i^sin #(1—5 cos 2 #) cos (<£ — &>+/')] 
dxf) 
Hence 
c 
H \e 
[3ir 
=\F cos + 3/J - \F' cos - +f 
= \F sin 3/-)- \F' sin f 
In the case of viscosity 
Therefore 
F— cos 3 f F'— cos f 
ur sin 6/4- 
JL 
1 (3 
Now if the obliquity had been neglected, the tidal friction due to the term of 
^ rj-2 
the first order in the tide-generating potential, would be given by y~ — ~i sin 46^ 
Hence 
1 / a\ V 5 sin 6/+ sin 2A 
i2i~ 8 \ c /\ sin4 ei / 
That is to say, this is the ratio of the terms neglected previously to those included. 
