200 P roceedings of ike Roydl Hociely of Victoria. 



<!ircnlar sections of the confocal (|uadiics Qi, Q2, Qs, ■ ■ • , 

 and such tliat the sum of the moments of inertia of the 

 entities ((i.il/j, a^. Mo, . . . , with respect to it, is of 

 constant magnitude 2.6', is a. (juadric cone of revolution. 

 Jiaving the point D^ as vertex, and IJ D as axis. 



We know that the locits of the lines 1 1 of intersection of 

 all pairs oi' mutually orthogonal tangent planes to any 

 quadric, cone G is another ([uadric, cone E concyclic with 

 the reciprocal of the cone C. (8ee Salmon's "Geometry of 

 Three Din:iensions," Art. 247). And if C be a cone, such 

 that the sum of the moments of inertia of the entities 

 ttj.iVi, «o. J/o, . . . , with respect to its tangent planes, 

 severally, be equal to a constant .s, we know that the sum 

 of the moments of inertia of the entities with respect to the 

 lines 1 1, severally, must be equal to 2.«. Hence we have : — 



Theorem 6. 



Given any masses Mj, Mo, Mg, . . . , in s{)ace and 

 corresponding numbers rtj, a.,, «;,, . . . , of known signs, as 

 multiplierh ; the Locus of a straight line II passing through 

 any given point Fin space, and such that the sum of the 

 moments of inertia of the entities rq. M^, a-j- Mo, a-^. M.^, .... 

 with respect to it — • any constant 2..s, is a quadric cone JE, 

 having the point V as ver-tex, and concyclic with the 

 reciprocal of the cone G, having V as veitex, and such that 

 the sum of the moments of inertia of the entities with 

 respect to its tangent planes = s, &c. 



Theorem 7. 



If three |)lanes, always mutually oi'thogonal, move in 

 space so as to continue to be tangent ])lanes respectively to 

 any three of the confocal quadrics Q^, Qo, Q:i', then will the 

 Locus of their common point of intersection be a Spltere, whose 

 centre is coincident with the mean centre of the entities 

 cij. i/i, ao. Mo, . . . , which is also the centre of the quadrics. 



Note. — This Theorem, which is an obvious deduction 

 from the kinetic properties exposed, was arrived at by 

 Salmon by means of a formula due to Chasles. (See 

 Sahnon's "Geometry of Tlnee Dimensions," Art. 172.) 



