308 
ME. GEOEGE H. DAEWIN ON THE INFLUENCE OF 
The parameter — 2 sin 2 y of II 1 is negative and numerically greater than the square 
of the modulus; therefore II 1 falls within Legendre’s second class (op. cit. tom. i. p. 72). 
Now it is shown by Legendre (tom. i. p. 138) that 
In this case & will be found to be y, ^ S . m , 5 =-jy \ / 2 . cos 2ry , and • 
2 /5 A (b,0) 2V cosy’ cosy’ 
whence a1=t-2{E 1 F-P(F-E)[, 
where the moduli of F and E are 8222 and their amplitude - — y. 
cos y 2 
From this form A may be calculated numerically. 
Appendix C. (Added April 1877.) 
Sir William Thomson, who was one of the referees requested by the Royal Society to 
report on this paper, has remarked that the subject of Part I. may also be treated in 
another manner. 
The following note contains his solution, but some slight alterations have been made 
in a few places. 
The axis of resultant moment of momentum remains invariable in space whatever 
change takes place in the distribution of the earth’s mass ; or, in other words, the normal 
to the invariable plane is not altered by internal changes in the earth. 
Now suppose a change to take place so slowly that the moment of momentum round 
any axis of the motion of any part of the earth relatively to any other part may be 
neglected compared to the resultant moment of momentum of the whole * ; or else suppose 
the change to take place by sudden starts, such as earthquakes. Then, on either suppo- 
sition (except during the critical times of the sudden changes, if any), the component 
angular velocities of the mass relatively to fixed axes, coinciding with the positions of 
its principal axes at any instant, may be written down at once from the ordinary formulae, 
in terms of the direction-cosines of the normal to the invariable plane with reference to 
these axes, and in terms of the moments of inertia round them, which are supposed to 
be known. 
Hence we find immediately the angular velocity and direction of the motion of that 
line of particles of the solid which at any instant coincides with the normal to the 
invariable plane at the origin. This is equal and opposite to the angular velocity with 
which we see the normal to the invariable plane travelling through the solid, if we, 
moving with the solid, look upon the solid as fixed. Let, at any instant, x, y, z be the 
direction-cosines of the normal to the invariable plane relatively to the principal axes ; 
* This is equivalent to neglecting |^ 2 , of Part I. ; by which Sir "W. Thomson is of opinion that 
nothing is practically lost. 
