460 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Jf A, B, C, D, E, F be the moments and products of inertia of the body about these 
axes of reference at any time ; H 1? H 3 , H g the moments of momentum of the motion 
of all the parts of the body relative to the axes; uq, on, oj 3 the component angular 
velocities of the axes about their instantaneous positions, the equations may be written 
— ( Aoq — F on — EonHj) -f- D (<u 3 3 — otq) -j- (C — B) onon ~b F ctnaq ~ Ectnoq 
4-w 3 H 3 — oj 3 H 3 =L.(23) 
and two other equations found from this by cyclical changes of letters and suffixes."" 
Now in the case to be considered here the axes A, B, C always occupy the average 
position of the same line of particles, and they move with very nearly an ordinary 
uniform precessional motion. Also the moments and products of inertia may be 
written A4-a 7 , 144“ 1/, C4-c', cl', e', f', where a', lb, c', d', e', f are small periodic 
functions of the time and a'd-b'd-c'^O, and where A, B, C are the principal moments 
of inertia of the undisturbed earth, so that B is equal to A. 
Now the quantities a', b', &c., have in effect been already determined, as may be 
shown as follows : By the ordinary formula! the force function of the moon’s action on 
the earth is 4~ r ^ + ^ + ^ — I j, where I is the moment of inertia of the earth about 
the line joining its centre to the moon, and is therefore 
— Ax 3 4~ By 3 4“ Cz 3 4” ar+ b'y 1 ff-c' 2 3 — 2d'yz —2 e'zx — 2f'xy. 
But the first three terms of I only give rise to the ordinary precessional couples, and 
a comparison of the last six with (11) and (13) shows that 
a' 
a 
IV 
b 
-. 0 . 
Also in the small terms we may ascribe to oq, on, cu 3 their uniform precessional values, 
viz. : oj } =— II cos n, on=— II sin n, oj. 3 =—n. 
When these values are substituted in (23), we get some small terms of the form 
a'n 3 sin n, and others of the form a'JTn sin n ; both these are very small compared to the 
terms in % and Jit—the fractions which express their relative magnitude being 
ID , Iffi 
— and 
T T 
There is also a term —nH, sin n, which I conceive may also be safely neglected, as 
also the similar terms in the second and third equations. 
It is easy, moreover, to show that according to the theories of the tidal motion 
of a homogeneous viscous spheroid given in the previous paper, and according to 
* Routh’s 1 Rigid Dynamics ’ (first edition only), p. 150, or my paper in the Phil. Trans. 1877, Vol. 167, 
p. 272. The original is in Liouville’s Journal, 2nd series, vol. iii., 1858, p. 1. 
t Routh’s ‘Rigid Dynamics,’ 1877, p. 495. 
