" Confocal Quadrics of Moments of Inertia." 207 



Theorem 8. 



If two planes A and B mutually orthogonal, be 

 tangent planes respectively to any two qiiadrics Q^, Q2, <»t' 

 the confocal family ; tlien will the other pair of tangent 

 planes A^ and B^ through their line of intersection II, to the 

 same two quadrics, l)e iinitually orthogonal. 



This is an obvious deduction from the kinetic properties 

 exposed. — The planes A and B being tangents to the quadrics 

 Qi and Q2, the moments of inertia of the entities d^. i/j, a^. 

 M2, ■ • • , with respect to them ai'e constants 6?! and So ; and 

 the sura Sj + Sj *^f these moments of inertia is equal to the 

 moment of inertia of the entities with respect to their line 

 of intersection 1 1. And since the moment of inertia with 

 respect to the line II m equal to the sum of the moments of 

 inertia with respect to the tangent planes A^ and B^, it 

 follows that A^ and B^ must be mutually orthogonal. 



This theorem is an e.x:tension to confocal quadrics of one 

 pertaining to confocal conies, due to Admiral De Jonquieres 

 of the French Navy, who is one of the most distinguished 

 geometers in Europe. (See "Melanges de Geomdtrie Pure," 

 par E. De Jonquieres.) 



Observations. 



The family of confocal quadrics Qi, Q2, Q^, • • • , and 

 the properties of inertia pertaining to them, are worth}' 

 of attention, not only on account of their intimate con- 

 nection with " Wave Surfaces," and " Surfaces of Elasticity," 

 but also on account of their direct applications to many 

 important problems. (See Salmon's " Geometry of Three 

 Dimensions," Arts. 467, 480, &c.) 



2°. — Some interesting properties pertaining to confocal 

 quadrics can be deduced by application of the numerous 

 new theorems arrived at by the author, and published in 

 Vol. X of the " Quarterly Journal of Pure and Applied 

 Mathematics," under the title — " Properties of Quadrics 

 having Common Intersection, and of Quadrics inscribed in 

 the same Developable." 



S". — Since writing the present paper, the author has 

 found that the question had been previously considered by 

 the late Professor Townsend, of the Dublin University. 



