202 Proceedings oj the Royal Society of Victo7ia. 



Duhamel's, as to the existence of two points, for each of 

 luhich Poinsot's " Ellipsoid of stress " is a Sphere. He 

 shows, moreover, these two points to belong to a "Focal 

 l-'onic" of the family of Confocal Quadrics. 



In the case in which the bodies are Spheres situated in 

 any manner in space, he gives a simple and effective 

 metliod of finding the tliree principal axes of inertia. 



He then records the followino- eioht Theorems, as results 

 01 his investigations : — 



Theorem 1. 



Given any raas.ses i/j, M^, M^, . . . in space, and 

 corresponding numbers a,, a.2, a^, ... of known signs 

 as multipliers. If a plane L L L (otherwise unrestricted) 

 be such that in every position it can assume, the sum of 

 the moments of inertia of the entities aj. M^, a^. M^, a^. M^, 

 . . . , with respect to it, be of any constant magnitude S, 

 then will the envelope of the plane be a determinable 

 (piadric Q, whose centre is coincident with the mean centre 

 of the entities. And the whole system of quadrics Qi, Q.2, Q^, 



corresponding to all values S,, S2, S^, . . . . , 



of S, will be concentric, coaxial, and confocal quadrics. And 

 in all cases in which the multipliers ai, a.^, . ■ ■ are all 

 positive, the quadrics will be Ellipsoids and Hyperboloids of 

 One Sheet. 



Theorem 2. 



Given any masses i/j, ill 2, M^, ... in space, and 

 corresponding numbers a^, a^, 0^3 , . . . of known signs, 

 as multipliers. The envelope of all planes LLL passing 

 through any given point V in space, and such that the sum 

 of the moments of inertia of the entities a^. M^, cio. M2, (tj. i/3, 



, with respect to them severally, is of any constant 



magnitude S, will be a determinable quadric cone C, 

 which envelopes a determinable quadric Q whose centre is 

 coincident with the mean centre of the entities. And the 

 whole family of such cones Cj, G.^, C3, . . . , corresponding 

 to all values 8-^, S2, S-^, . . . , of S, will be coaxial and 

 confocal cones enveloping coaxial and confocal quadrics, 

 whose common centre is the mean centre of the entities 



