456 MR. G. H. DARWIN ON THE PRECESSION OE A VISCOUS SPHEROID, 
these there are three couples, ?i, ffX . suppose, caused by the attraction on the 
wave surface o\ 
As it is only desired to determine the corrections to the ordinary theory of preces¬ 
sion, the former may be omitted from consideration, and the attention confined to the 
determination of %, J$l, Jl. 
The moon will be treated as an attractive particle of mass to. 
Now cr as defined by (9) is a surface harmonic of the second order; hence by the 
ordinary formula in the theory of the potential, the gravitation potential of the tide- 
wave at a point whose coordinates referred to A, B, C are ry, r£ is f- 7 rwci(^j a or 
JTa 
f —rcr. Hence the moments about the axes A, B, C of the forces which act on a 
particle of mass to, situated at that point, are 
? mMa( da f<r\ 
, &c., &c. 
Then if this 
particle has the mass of the moon; if r be put equal to c, the moon’s distance; and if 
7], £ be replaced in cr by x, y, z (the moon’s direction cosines) in the previous expres¬ 
sions, it is clear that — &c., &c., are the couples on the earth caused 
by the moon’s attraction. 
These reactive couples are the required %, JJift, Jl- 
Hence referring back to ( 12 ) and remarking that -§Ifcr=C, the earth’s moment of 
inertia, we see at once that 
f =‘“[( c ■“%z—d(r-s 2 ) - exyf Hr] 
[(a— c)zx— e(z 3 —ar) — fys+daq/] > 
t' 9 
ID 2t 2 
77 = — [ (b - a) xy-- f(x 3 ■- y°) - d zx -+ eyz] 
L 9 J 
(13) 
Where the quantities on the right-hand side are defined by the thirteen equations 
( 7 ) and ( 10 ). 
I shall confine my attention to determining the alteration in the uniform precession, 
the change in the obliquity of the ecliptic, and the tidal friction; because the nutations 
produced by the tidal motion will be so small as to possess no interest. 
In developing % and Jit I shall only take into consideration the terms with argu¬ 
ment n, and in J-l only constant terms ; for it will be seen, when we come to the 
equations of motion, that these are the only terms which can lead to the desired end. 
§ 4. Development of the cowples % and jm. 
Now substitute from (7) and ( 10 ) in the first of (13), and we have 
11 
C 
0 
