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PROCEEDINGS OP THE AMERICAN ACADEMY 



But dQ is in the direction of the variation of q at any instant. It is 

 then in the direction of the tangent, at the extremity of q (Fig. 1^ 

 (Tait, § 36). 



Fig. 2. 



Now if we consider q fixed, but allow do to vary, we may write (co — n) 

 for dQ (Fig. 2), and the equation 



Sdn(pQ = S(a) — Q)q)Q = 



is that of a plane containing any tangent do ; and, therefore, of the 

 tangent plane. Since the extremity of q is on the surface, we have 



HqcPq = 1, 



and the equation of the tangent plane may be written 



SacpQ = 1. 



We see (Tait, § 205) that qpp is perpendicular to this surface, or, in 

 other words, in the direction of the normal ; and if we take co in the 

 direction of qjQ, or = arqpp, 



SxqjQCpQ = x(q!Qy = 1 ; 



*^^ ^p 





and this last is the perpendicular from the centre on the tangent plane. 



Conjugate Diameteus and Diametral Planes. 



If we want to find a line co through the origin which bisects all 

 chords parallel to another line «, (w -|- xa) and (w — xa) must both 

 terminate in the surface : that is, w must satisfy the equation 



S(ft) ± a;a)fjp(a) ± xa) =. 1, 



where a; is a scalar. Now, if we develop this equation, we find 



Scoqpco -\- x'Saqpa ±, 2 x Scoqja = 1. 



