202 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
gives the velocity of the point {J) hi space, the differential letter, d, denoting ahsolute 
change of position. But it is often important to determine the motion of this point 
relatively to the rigid body ; and this may be done as follows : — 
Let us assume S to denote relative change of position with reference to the rigid 
body ; then it is evident that 
dw' 8co' _ , 
( 11 -) 
for the velocity of the point {a) is the resultant of two velocities, namely, that relative 
to the body, and that arising from the motion of the body; the former is and the 
latter, by ( 9 .), is 'Da.oJ. 
Hence, and by (10.), we find 
( 12 .) 
This equation gives the velocity of the point {u) relatively to the rigid body. 
Now it is clear that if we can solve (10.) and (12.) the motion of the rigid body is 
determined ; for we shall then know (by (12.)) how the line u' moves in the rigid body, 
and by (10) how cj' moves in space-, and thus, by the intervention of oJ, we shall 
obtain the motion of the rigid body in space. 
(107.) As an example of this I shall take the case of the earth attracted by the 
sun, and point out briefly how (10.) and (12.) determine the motion of the polar axis. 
In this case A=B, and C=(l+?i)A, when>^isasmall number: also the instantaneous 
axis CO very nearly coincides in direction with the polar axis 7 . Hence, and by art. 
89 , we have 
iy' = A (iy T" "kco^y) 
. U= Ay^Dw' . (w'+^^'y). 
Here n! denotes (symbolically) the sun’s distance, d is the magnitude of m' the 
sun’s mass. 
Thus, observing that Dm'.m' and J}co.co are zero, (10.) and (12.) become 
y 
dco Sfwov) 3m' , 
dt 
■'kco.J^co .y. 
In the terms multiplied by \ we may approximate on the supposition that 7 is fixed 
and co—ny, where n is the earth’s angular velocity about its polar axis. This reduces 
the two equations to 
dm 3m! . 
.y. ( 13 .) 
