On the Attraction of Ellipsoids. 361 



PROPOSITION V. 



A. system of material particles attracts a point M, whose distance 

 from the centre of gravity of the attracting system is very great 

 compared with the mutual distances of the particles ; then if a tan- 

 gent plane be drawn to the " ellipsoid of gyration ,"* perpendicular 

 to OM, the whole attraction lies in the plane OST, where S is the 

 point in which this tangent plane intersects OM, and T its point 

 of contact with the ellipsoid. 



Let a, /3, 7 be the direction angles of OT ; a', /3', 7' of OM ; 

 and ai, /3i, 71 of TS ; and a , /3 , 70 of the normal to the 



\M 



Fig. 4. 



plane OST ; and let OS, OT and the angle SOT, be denoted by 

 p, r, and respectively. It will be sufficient to prove that the 

 component Q of the attraction in the direction of the normal to 

 the plane OST is zero. 



We shall first find the components X, Y, Z, of the attraction 

 in the directions of the axes, and thence deduce the value of Q. 



* The centre of this ellipsoid is at the centre of gravity ; its axes are in the 

 directions of the principal axes, and their lengths are determined hy the equations 



Ma* = A, MV t = B, M<? = C. 



This ellipsoid is used hy Professor Mac Cullagh in his " Theory of Rotation": see 

 Rev. S. Haughton's Account of Professor Mac Cullagh's Lectures on that subject, 

 Transactions R. I. A., VOL. xxu. p. 139 {supra, p. 329). 



