274 
ME. GEOEGE H. DAEWIN ON THE INFLUENCE OE 
“ Since the fixed axes coincide with these at the time t, we have x=od, y=y', and by 
Art. 114, 
dx dad * . "'I 
d l— H .a »_ a , i’ 
dt~dt+®* X &lZ J 
and by similar reasoning 
■. tf.=H 8 + C0 3 -E0 I -D0 2 *; 
K= H.+M-Ffl.-Ed,, 
h' 2 =n 2 + bq 2 -d&;-fq 1 . 
“ Hence the general equation of motion becomes 
j t (C« J -E« 1 -D« s +H s )+F(9i-fl?)+(B-A)flA+E«A-D9A+4,H 2 -fl J H,=N . (7) 
and two similar equations. 
“ Let the moving axes be so chosen as to coincide with the principal axes at the 
time t. Then D = 0, E=0, F=0, and the equations become,” 
and two similar equations ; where X 2 , X 3 (replacing the A, B, C of Mr. Routh) are 
the three principal moments of inertia, and are functions of the time. 
In order to apply these equations to the present problem, we must consider the 
meaning of the quantities fl 2 , fi 3 . A system of particles may be made to pass from 
any one configuration to any other by means of the rotation of the system as a whole 
about any axis through any angle, and a subsequent displacement of every particle in a 
straight line to its ultimate position. Of all the axes and all the angles about and 
through which the preliminary rotation may be made, there is one such that the sum 
of the squares of the subsequent paths is a minimum. By analogy with the method of 
least squares this rotation may be said to be that which most nearly represents the 
passage of the system from one configuration to the other. If the two configurations 
differ by little from one another, and if the best representative rotation be such that 
the curvilinear path of any particle is large compared to its subsequent straight path, 
the system may be said to be rotating as a rigid body, and at the same time slowly 
changing its shape. Now this is the case we have to consider in a slow distortion of 
the earth. 
Divide the time into a number of equal small intervals r, and in the first interval let 
the earth be rigid, and let each pair of its principal axes rotate about the third (with 
angular velocities double those with which they actually rotate). At the end of that 
interval suppose that each pair has rotated about the third through angles 2 &> L r , 2 v. 2 t, 
* A, B, C, D, E, E are, as usual, the moments and products of inertia. 
