" ConfoccU Quadrics of Moments of Inertia." 203 



«!. Ml, do. M.2, And if the point V be at infinity, 



unci given in direction by means of a vector R passing 

 thi'ough the mean centre ; then, corresponding to various 

 values of S, the envelopes oi L L L consist of a system of 

 confocal cylinders enveloping the quadrics, dnd having as 

 common principal axis the directing vector R. 



Now Ml, M.2, M^, . . . being masses, and a^, a^, a^, 

 numbers known in signs: we know that if a plane ZX 2/ 

 l)e such that the sum of the moments of inertia of the 

 entities 11^. M^, cu M^, a-^. Mo^, . . . , with respect to it is of 

 a constant magnitude S, then will the envelope of the plane 

 be a determinable quadric Q. But the line of intersection 

 / I of any two mutnalh' orthogonal planes, both tangent 

 to the quadric Q, is obviously such that the sum of the 

 moments of inertia of the entities with respect to it is 

 represented by 2..s'. 



We can easily form the equations of tangent planes 

 to tile quadric Q, and express their mutual orthogonism ; 

 but we need not try to evolve an equation of a surface 

 which could be the envelope of all the lines 1 1 of intersection 

 of the pairs of mutuall}' orthogonal tangent planes to Q. 

 This is obvious : — for if we suppose j:) to be any point 

 whatever on any surface, and construct a Poinsot Elli))soid 

 having such point as centre, we perceive that the lines / 1 

 tlirough the point form a cone, and cannot generally ell be 

 tangents at one point to any other surface. However, we 

 j)roceed to find the Loci and Envelo})es of lines ly l^ which 

 fulfil the conditions as to equality of moments of inertia, 

 and respecting which other conditions are imposed. 



1°. — With respect to all the lines li iy wliich are parallel 

 to any fixed straight line R R passing through the mean 

 centre 0, which is also the centre of the quadric Qj. 



If throLigh we draw a plane normal to the line R R, 

 and that we put c^ c^ Cy to represent the conic which con- 

 .-stitutes its trace on the quadric Qi : then, from a well-known 

 theorem, we perceive that the pairs of mutually orthogonal 

 tangent planes whose points of contact lie in the conic 

 ''iCiC,,give us all the lines lilx parallel to the fixed line 

 RR, and that they constitute a Right Circular Cylinder 

 liavinw R R as central axis. 



