" Confocal Quadrics of Moments of Inertia^'' 2().> 



the projections of the ([uadiMcs on the pLme be ;i system of 

 concentric circles. 



Note. — The differences of the moments of inertia with 

 respect to the lines ^j/j, /., /o- ^;5^3> • • • . (tangents to the 

 respective conies) on the plane BBB are obvionsly equals to 

 the differences of the moments of inertia with respect to 

 tangent planes to the qnadrics Qi, Q2, Qs, ■ • ■ 



If we draw planes Pi Pi Pi, P2P2P2) ■ ■ ■ , tiirough any 

 diameter D D of any one Q of the family of Confocal quadrics. 

 the lines / 1 situated in these ])lanes and sucli that the sum 

 of the moments of inertia of the entities %. J/i, «o. M.y, '^3. il/;j, 

 .... with respect to them, several!}'-, is of any constant 

 magnitude 2.s, have (as alrendy observed) as envelopes, in 

 the ]ilanes, determinable conies. And we know that those 

 of the lines II which are parallel to D D form a circular 

 cylinder; having the line D B as axis. But it is easy to 

 ])erceive that it is onl}' when the axis D D is normal to one 

 of the circular sections of the quadric Q that the conies cut 

 D D in the one and same point, at which the lines 1 1 form a 

 tangent plane to all the conies. Hence : — 



Theorem 4. 



Given any number of masses i/i, J/o' -^^3' • • • > in space, 

 and corresponding nundjers a^, cio, a^, . . . , of known signs 

 as multipliers ; if a straight line 1 1 move in space so as to be 

 always in contact with the line D i) of a diameter of any 

 quadric Q (of the confocal family) normal to either system of 

 its circular sections, and so that in every position the sum 

 of the moments of inertia of the entities aj. il/j, «o. i/o, . . . , 

 with respect to it, is of any constant magnitude 2.s ; then 

 will the envelope of the straight line Z ^ be a determinable 

 quadric lu of revolution, having the mean centre as centre, 

 and the fixed line B B as axis. And all such quadrics 

 '"1, 10-2, iv-i, . . . , corresponding to all possible values 

 2.S1, 2.S.2, 2.S3, . . . , of the constant are determinable 

 quadrics of revolution, having the mean centre as common 

 centre, and the line Z) Z> as principal axis. 



Theorem 5. 



The Locus of a straight line 1 1 through any fixed point B' 

 in a line B B through the mean centre and normal to 



