544 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
Let XYZ (fig. 1) be rectangular axes fixed in the earth, Z being the axis of 
rotation and XZ the plane from which longitudes are measured. 
Let M be the projection of the moon on the equator, and let oj be the earth’s 
angular velocity of rotation relatively to the moon. 
Let A be the major axis of the tidal ellipsoid. 
Let AX=a>L where t is the time, and let MA=e. 
Let m be the moon’s mass measured astronomically, and c her distance, and 
Then according to the usual formula, the moon’s tide-generating potential is 
rr 2 [sin 3 9 cos 3 (<j> — cot — e) — -j], 
which may be written 
-|rr 3 (-g-— cos 2 9)-\-^rr 2 sin 2 9 cos 2(<£ — cot — e). 
The former of these terms is not a function of the time, and its effect is to cause a 
permanent small increase of ellipticity of figure of the earth, which may be neglected. 
We are thus left with 
sin 2 6 cos 2 [(f) —cot—e) 
as the true tide-generating potential. 
Now if tan 2e= * —, where v is the coefficient of viscosity of the spheroid, then by 
rjcao J 
the theory of the paper on “ Tides,” such a potential will raise a tide expressed by 
~ = h~ cos sin 3 9 cos 2 (<]) — a)t)* .(2) 
Then if we put 
S = 4t sin 2 6 cos 2 (<j> — (ot — e) .(3) 
S— sin 2e sin 2 9 sin 2 (<j) — ojt) .(4) 
and 
—— = r sin 2e sin 9 cos 9 sin 2 ((f)—cot) 
she 4 ( s -*3= t siu 2e siu 9 cos 
Multiplying these by m;« 3 -, we find from (1) the tangential stresses communicated 
by the layer cr to the sphere. 
* “ Tides,” Section 5. 
