LIQUID SPHEROID OF FINITE ELLIPTICITY. 
217 
where, in this case, 
z=: ^5 cos a (1 — sin^ e)~^ . . .... (105), 
the corresponding value of k being Is. 
Equations (102), (103), (104) fully determine the height of the forced tides and 
deformations on the surface of the spheroid due to the attracting body. 
If, instead of one attracting mass m, we suppose two equal masses on opposite 
sides of the body, the expression for Vg, and therefore for Ji, will contain only zonal 
harmonics, and harmonics of even rank, but these will be the same as before. 
Harmonic Tides of the Second Order. 
27. If the body be very distant from the spheroid, Z will be great, and the series 
(101) will converge very rapidly, so that the harmonics of the second degree are the 
most important. 
Taking n = 2, s = 2, we get 
p 3_^ (0 V(Z) 
- 12 M ‘ Ko^ (?) - {21 - 1) q, (?) 
(106), 
and the corresponding term in the height of the forced tide is 
h. 
3 
™ (?- + l)w/(Z) 
4 MK/(?)-4((2^-l)^„(?)'^ 
— fT) <77 cos 2 [(f) 2iii wt) 
(107). 
If the attracting body be rotating in the same direction as the liquid, but with less 
angular velocity (as in all cases of astronomical interest), 2l lies between 0 and 1. If, 
in addition, Kf (?) is positive, the denominator in the expression forcannot vanish, 
and the tide produced by this term cannot therefore become very large. In fact, the 
angular velocities of both the free harmonic tides will lie beyond the above-mentioned 
limits, and will neither of them coincide with that of the attracting body. If, 
however, (?) = 0, and if, in addition, the attracting body be fixed in space, so that 
2l = 1, this forced tide will increase indefinitely. The spheroid will now have a semi¬ 
diurnal free tide, which will be fixed in space, and will coincide with the forced tide 
clue to the attracting body. The equilibrium in the spheroidal form will therefore be 
completely broken up. 
If the attracting body be rotating very slowly about the spheroid, the same thing 
will happen if (?) has a certain corresponding small negative value. 
Since the spheroid is secularly stable or unstable according as Ko^ (?) is positive or 
negative, the mode in which its relative equilibrium will be destroyed if (?) 
becomes negative will depend on circumstances. If the liquid possess but little 
MDCCCLXXXIX.-A. 2 F 
