AND ON THE REMOTE HISTORY OF THE EARTH. 
513 
§20. Terms of the second order in the tide-generating potential. 
The whole of the previous investigation has been conducted on the hypothesis that 
the tide-generating potential, estimated per unit volume of the earth’s mass, is 
tCTr 2 (cos 2 PM— but in fact this expression is only the first term of an infinite 
series. I shall now show what kind of quantities have been neglected by this treat¬ 
ment. According to the ordinary theory, the next term of the tide-generating 
potential is 
Y z =w™(£j °(f cos 3 PM-f cos PM) 
Although for my own satisfaction I have completely developed the influence of this 
term in a similar way to that exhibited at the beginning of this paper, yet it does not 
seem worth while to give so long a piece of algebra ; and I shall here confine myself to 
the consideration of the terms which will arise in the tidal friction from this term in 
the potential, when the obliquity is neglected. A comparison of the result with the 
value of the tidal friction, as already obtained, will afford the requisite information as 
to what has been neglected. 
Now when the obliquity is put zero (see Plate 36, fig. 1), 
cos PM=sin 6 sin(<£ — co) 
where co is written for n — fl for brevity. Then 
cos 3 PM=f sin 3 6 sin (</> — co) —^ sin 3 6 sin 3(<£ — co) 
and 
cos 3 PM—f cos PM=t 2 - 0 - sin 0(1—5 cos 3 6) sin ( cf)—a >)—^ sin 3 0 sin 3(<£ — &>). 
Then since 
m/r\ 3 5 r 3 5 
V)- - o = WT- - 
c\c] 2 c 3 
therefore 
Vo -Mt>-r 3 = — -^sin 3 0 sin 3(<£ — co) + -j sin 0(1 — 5 cos 2 0) sin (f> — co) 
If sin 3 ((f) —co) and sin (<£ — co) be expanded, we have V a in the desired form, viz. : a 
series of solid harmonics of the third degree, each multiplied by a simple time har¬ 
monic. Now if rrr 3 S 3 cos (yt-^-rj) be a tide-generating potential, estimated per unit 
volume of a homogeneous perfectly fluid spheroid of density w, S 3 being a surface har¬ 
monic of the third order, then the equilibrium tide due to this potential is given by 
cr=^— S 3 cos (vt + rj), or a = S 3 cos (vt-f rj). Hence just as in Section 2, the tide- 
* See Section 1. 
3 U 2 
