Professor Mac Cdllagh's Lectures on the Attraction of Ellipsoids. 387 



^" ? + 27"^ ^ ('^''' + yy' + ^^'^' dm-^,^{x^ + f + z^) dm, 



x' y' z' being the co-ordinates of the distant point, and r' its distance from the 

 origin. Let now the principal axes at that centre be taken as axes of co-ordi- 

 nates ; then, since 



'Sixydm = 0, "Exzdm = 0, "Zyzdm = ; 



^= 7 + 27-' ^ ^'^'■'^" + y'y" + ^'^')'^"' - 27? S (^^ + 2/= + z') dm. 



Hence, if ^, B, C be the three principal moments of inertia, and / the mo- 

 ment of inertia round OM, 



y=y^^A^+B^C-U). 



(6) 



Proposition V. 

 A system of material 

 particles attract a point , 



M, whose distance from 

 the centre of gravity O of 

 the attracting mass 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 

 plane OST, where S is the point in which this tangent plane intersects OM, 

 the point of contact. 



Fig. 4. 



\M 



in the 

 andT 



* 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 by the equations 



Ma' = A, MV = B, Mc' = C 

 This ellipsoid is used by Professor Mac Cullagh in his Theory of Eolation ; see Rev. S. 

 Hacghton's Account of Professor Mac Ccllagb's Lectures on that subject, Transactions E. I. A. 

 vol. xxii. p. 139. 



VOL. XXII. 3 B 



