WITH THE TIDES OF A VISCOUS SPHEROID. 
553 
we should, therefore, have to equibrate the system by bodily forces arising from the 
effects of the inertia due to the uniform precession and the fortnightly nutation—-just 
as was done above with the tidal friction. This would be an exceedingly laborious 
process; and although it seems certain that the tides thus raised would be very small, 
yet we are fortunately able to satisfy ourselves of the fact more rigorously. Certain 
parts of the tangential stresses do form an equibrating system of forces, and these are 
precisely those parts of the stresses which are the most important, because they do not 
involve the sine of the obliquity. 
I shall therefore evaluate the tangential stresses when the obliquity is zero. 
The complete potential due both to the moon and to the diurnal rotation is 
r 2 S=-|r 2 (w 3 +T)(|-— cos 3 9) +-|r 3 T sin 3 9 cos 2 ((f) — cot —e), 
and the complete expression for the surface of the spheroid is given by 
5 =i(n 3 d-r)(|-— cos 3 6) + t 2 t cos 2e sin 2 6 cos 2 ((f) —cot). 
He] ice 
S — Qy =-|rsin 2e sin 2 9 sin 2 ((f)—(ot). 
Then neglecting r 3 compared with rn 3 , and omitting the terms which were previously 
considered as giving rise to secular distortion, we find 
wa ~7, S — tjy)= ivcPt^-- sin 2e sin 9 cos 9(\— cos 2 9) sin 2 ((f) —cot), 
wa 
a cW\ 
a cl 
a 
^edd>\ s ~K ) =w, « ?T H- sil12e sin #(i- cos °~ e ) cos 2(<£-<»0- 
The former gives the tangential stress along, and the latter perpendicular to, the 
meridian. 
If we put e=-l—, the ellipticity of the spheroid, we see that the intensity of the 
tangential stresses is estimated by the quantity wed .to sin 2e. But we must now find 
a standard of comparison, in order to see what height of tide such stresses would be 
competent to produce. 
It appears from a comparison of equations (7) and (8) of Section 2 of the paper on 
“ Tides,” that a, surface traction S» (a surface harmonic) everywhere normal to the sphere 
produces the same state of flow as that caused by a bodily force, whose potential per 
unit volume is fMs,-; and conversely a potential W, is mechanically equivalent to a 
surface traction 
Now the tides of the first order are those due to an effective potential wi 
4 b 2 
