" Oonfocal Qiiadrics of Moments of Inertia. 



201 



envelo|)e of the plane L L L is a Quadric whose centre is 

 coincident with the mean centre of the points jPj, P-i, P^, 



. . . , and their respective multipliers «i, a2, a^, 



And he shows that tlie quadrics corresponding to all 

 possible values of the entity ^.S, are Confocal Quadrics. 



In order to amplify his Geometrical Method, he proceeds 

 to give a full and complete solution to the particular cases 

 in which the given points Pj, Po, Pg, . . . , are all in one 

 straight line. And he shows that it depends on the state of 

 the data, as to whether the Confocal Quadrics be Ellipsoids ; 

 Hyperboloids of One Sheet ; Hyperboloids of Two Sheets ; 

 Spheres ; or Paraboloids. 



He then directs attention to the Physical Aspect of the 

 problem, which he enunciates as follows : — 



Problem. — Given any masses M^, M2, i/3, . . . , in space, 

 and corresponding units a-^,a.2,a.i, ■ ■ . , known in signs as 

 their respective multipliers; to find the Envelope of a 

 plane L L L, such that in every position it can assume, 

 we shall have the sum of the Moments of Inertia of the 

 masses represented by 



a, . ^ m, (Pi Lf + a.2 . ^ m. (P^ Lf + a,.^ m, . (P3 Lf 

 ■+... = a constant S, 



in which 97ii, liiaj '>''*'3, • • • represent molecules of the masses 

 i/i.jyo.Jfg, . . . , at any points P^, P2, P3, ... in those 

 masses, and in which P^L, PoZ, P^L, . . . represent the 

 pedals from the points Pj, Po, P,, . . . , to the plane LL L. 



In elucidation of this aspect of the problem, he recon- 

 siders the particular cases, in which he now replaces the 

 given points or molecules at Pj, P2, P3, ... all in one 

 line, by Spheres whose centres are all in one straight line. 

 He shows that the results arrived at previously, apply 

 when masses replace mere molecules ; and that, according 

 to analogous states of the data, the Confocal Quadi-ics will 

 be Ellipsoids, Hyperboloids, Spheres, or Paraboloids. 



He establishes the limiting values for the constant S, and 

 exposes the limiting forms of the Quadrics in minute and 

 full detail. And he corroborates a remarkable theorem of 



