WITH THE TIDES OF A VISCOUS SPHEROID. 581 
approach the critical point where n cos i= 2fl, where the rate of change of obliquity 
was found to vanish, the tidal movements might have become so rapid as seriously to 
affect the correctness of the tidal theory used ; and accordingly it might be thought 
that the critical point was not reached even approximately when n cos i=2fi. 
The preceding analysis will show at once that this is not the case. Near the critical 
point the solar terms have become negligeable; then if we put r =0 in the first of 
equations (G9) we have 
siD 4e -i^[ ] — 2Xsec * + K(l — 2\)(1 — 
The condition for the critical point in the first approximation was 2\ sec i— 1 ; if 
then i is so small that we may take sec i— 1 in the inertia term, this condition also 
causes the inertia term to vanish. 
Hence the corrected theory of tides makes no sensible difference in the critical point 
cli 
where — changes sign. 
Having now disposed of these special points connected with previous results, I 
shall return to questions of general dynamics connected with the approximate solution 
of the forced vibrations of viscous spheroids ; that is to say, I shall compare the results 
with those of— 
The forced oscillations of fluid spheroids.* 
The same notation as before will serve again, and the equations of motion are 
dp , d W da. „ q 
Wfc + &-“V =0 j 
two similar equations }> . 
(73) 
i il,J - , ^3 . <h __ . 
and * + ,7,, + A- 0 J 
If the external tide-generating forces be those due to a potential per unit volume 
equal to tor’S t , and r=«-b<x, be the equation to the tidal spheroid, where S;, cr t are 
surface harmonics of the i th order, then we must put 
the second term being the potential of the tidal protuberance, and the last of the mean 
sphere. 
Differentiate the three equations of motion by x, y, z and add them, and we have 
V ~(p- w( 3« 2 - >*)£) — 0. 
* This is a slight modification of Sir W. Thomson’s investigation of the free oscillations of fluid 
spheres, Phil. Trans., 1863, p. 608. 
