OP ARTS AND SCIENCES. 227 



But this is evidently impossible, unless 



Smcpa = 0. 



The locus of co is, then, a plane perpendicular to qpoc and passing 

 through the origin. It is also parallel to 



SQq)a = 1, 



which is the tangent plane at the extremity of a vector through the 

 origin parallel to a. Conversely, if 



Saxfa = 0, 



the locus of CO is a plane bisecting all chords parallel to «, because 

 some X, a scalar function of co, can evidently be found, such that 



S(a) ± xa)cp((o ± xcc) = 1. 



Since go is self-conjugate, we have 



Saqico = Sco(jp« = 0, 



so that the relation is reciprocal ; and if co be constant, and a vary, 

 the locus of the latter is a plane parallel to the tangent plane at the 

 extremity of a diameter parallel to co. If ^ is any vector lying in the 

 first plane, our two planes will be denoted by 



Scocfa = 

 and 



Sco(jp'(3 = 0, 

 and we have 



Sag}|3 = S|?(jp« = 0. 



The intersection of the two planes is Ycfiacf^, because this satisfies both 

 equations : for 



S^aqp^cprc = = Scfcicp^cp^. 

 And, denoting this intersection by y, we see that 



Saqpy = Sycfa = 0, 

 and 



S§cpy = Sycp^ = 0. 

 Now 



Scogjy = 



