Art. XVII. — On " Confocal Quadrics of MomenU of 

 Inertia " 'pertainiiig to all Planes in Space, and 

 Loci and Envelopes of Straight Lines vjhose 

 " Moments of Inci-tia " are Constant. 



By Martin Gardiner, C.E. 



iRead May 12, 1892.] 



Abstract. 



The author commences by solving the following problem, 

 by the Cartesian co-ordinate method : — 



Problem. — Given any number of points Pj, P^, P^, .... 

 in space, and corresponding numbers «!, a^, «3, . . . , known 

 in signs and magnitudes as respective multipliers ; to hud 

 the Envelope of a plane L L X, such that, in every position 

 it can assume, we shall have 



a^ .p\ + a^.pl + (^3 . pi -\- . . . . = 8, 



in which p\,pl,'pl, . . . . , represent the squares of the 

 pedals from the points Pj, Po, P3, . . . , to the plane L L L, 

 and S a constant entity known in sign and magnitude. 



He finds the equatloii of the envelope of the plane L L L 

 to be that of a Quadric whose centre is coincident witii 

 the meaTi-centre of the given points for the multipliers 

 «i, a^, a-i, . . . And from the form of the equation arrived 

 at (which is given abridged and expanded), he infers that 

 for all possible values of the entity S, the corresponding 

 Quadrics are Confocal Quadrics. 



He then shows by a purely geometrical method (indepen- 

 dent of co-ordinates) that for an}^ constant value of *S', the 



