192 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
Hence, observing that u.u—^, and 2mM=0, we have 
2w, . U — Xu ){ u . w') 
3m' , ^ , 
= —^v!.{Zm{u Xu)u}. 
Now '^ni{u' X u)u='S,m{xj(^ -{-yy' -\-zz')(a:u-^y(3-\-zy) 
— {'l,mj!:^)x'a-\-(2my^)y'^-{-{2mz^)z''y, by properties of principal axes, 
= {'2mr^)u'— {Ax’a-\-By'^-\-Cz'y). 
Wherefore, observing that m'.m' = 0, we find 
3m' 
. U =—f^u' . (Ax'd + By'j3 + Cz'y). 
Also 
2U=2?wm' (neglecting " fof obvious reasons) 
3w' X u 
Hence we have 
rM = 2??z 
[2?wm=0. 
Mortua = Mm' ps-j-'-p^u ' . {Ax'a-\-By'^-\-Cz’y), 
which expresses a force Mm' p^ acting at the origin, and a couple, 
^TYi^ •» 
-p^id . {Ax'cc-YBy'^-{-Cz!y). 
If we perform the lateral multiplication indicated by {v!.), this couple becomes 
-pr|(C-B)y2'/3.r+(A-C)xVy.»+(B-A>y».|3L 
which gives the three well-known couples in the theory of Precession and Nutation. 
IV. Application of the Symbolic Forms to determine the Correction for the 
Earth’s Rotation in Problems relating to motion on or near the Earth's 
Surface. 
(90.) The best way of defining that vdiich is commonly called the Centrifugal Force 
appears to be the following, viz. that it is an imaginary force which may be introduced 
as a correction for the error of not taking into account the rotation of the radius 
vector r. Suppose P to denote the accelerating force acting along r, and let us fora 
moment forget that r has an angular velocity then we put 
but this is erroneous, and we must correct it for the rotation by adding to P the 
term r y-p-J , as is M^ell known. Hence we may regard the centrifugal force as a cor- 
rection for neglected rotation. 
