OF ARTS AND SCIENCES. 231 



Poles and Polar Planes. 



To find the locus of the point of harmonic division of radii through 

 a point «, transfer the origin to that point, and the equation 



S()q)Q = 1 

 becomes 



SQqiQ -\- 2 S()qp« -j- Sacpa ■= 1. 



Let the vectors of the surface on any line passing through the new 

 origin be q' and (/', and let their harmonic mean be q. We must have 

 then 



Tp Tp> ^ Tp" 



If, now, we take q in the direction ^, the equation of the surface 



becomes 



TQ^S^cf^ + 2TQ^^cfa + Sa(pa = 1. 



Tq' and Tq" are the roots of this quadratic ; and applying to it the 

 well-known principles of quadratics, we have 



2 1 I 1 Tp" + Tp' — 2 S;3<pa . Sa<pa — 1 — 2 S0<pa 



¥p Ty T^ Tp'Tp" S^^j3 '• S0(p0 Sac^o — 1 ' 



which gives 



TpS/3g)« =1 — S«g)a; 



or, since ^ is the versor of p, 



Spqpa -\- S«qDcc = 1 ; 



which is the equation of a plane, the polar plane of the new origin. 

 Transfer back to the former origin, by substituting q — a for q, and 

 we find 



Spgja = 1 



as the equation of the polar plane of a. 

 This last equation can be written 



Saq)Q = 1, 



and by changing the variable we get, of course, the polar plane of q. 

 So we see that, if a is on the polar plane of /3 (Saq)^ = 1), |3 is on the 

 polar plane of a (Sj^(jp« = 1). 



We have found that the equation of the tangent plane at a point q is 



Sooqp^ = 1. 



