WITH THE TIDES OF A VISCOUS SPHEROID. 
557 
In a later investigation we shall require a transformation of the expression for —, 
and as it will here facilitate the integration, it will be more convenient to effect the 
transformation now. 
If (+, Qi. be the zonal harmonics of the second and fourth order, 
cos- d—^Qo + 
cos 1 6— 3^5 Qj,+yQ 2 + w 
Now 
( 8 a 2 — 5r 2 ) 2 —-fr- sin 2 6[S 2a~ — (26+ sin 2 d)r 2 ] 
= (8cr — 5r 2 ) 2 — r 2 [48a 2 —++ — f-(32a 2 —28+) cos 2 6 —fr 2 cos 1 6 \ 
= i{320+-560aV 2 + 259+}-f(112a 2 -95r 3 )r 2 Q ;J +i|7’ 4 Q i . . . (27) 
The last transformation being found by substituting for cos 2 6 and cos 4 6 in terms of 
Q., and (+, and rearranging the terms. 
The integrals of Q > and Qvanish when taken all round the sphere, and 
i / 
(3 20a 4 — 5 6 0a 2 r 2 + 2 5 9r 4 ) r 2 sin 0dnl0dcf>=^{^-^+^} = X 19, 
where C is the earth’s moment of inertia, and therefore equal to -+ 7 nucd, 
Hence we have 
Iff 
sin 6drddd<b=- 
dt T v 
\ sin X 19 = +(r sin 2e) ! C. 
„y / " dov 
-D n n r, r, ^ VU > , 1 , ‘ JWCr M , o 
nut tan 2e =—- = 2 .--so that cob 2 e. 
yaw 5 Qioa~ oov a 
T~ 
And the whole work done on the sphere per unit time is sin 4e.(Ja». 
Now, as shown in the first part (equation 5), if be the tidal frictional couple 
V=ff sln 4e - 
Therefore the work done on the sphere per unit time is $£Ig). 
It is worth mentioning, in passing, that if the integral be taken from \a to 0 , we find 
. that "32 of the whole heat is generated within the central eighth of the volume ; and 
by taking the integral from \a to a, we find that one-tenth of the whole heat is generated 
within 500 miles of the surface. 
It remains to show the identity of this remarkably simple result, for the whole work 
done on the sphere, with that used in the paper on “ Precession.” It was there shown 
* Todhunteb’s ‘ Functions of Laplace,’ &c., p. 13 ; or any other woi’k on the subject. 
