228 PROgEEDINGS OF THE AMERICAN ACADEMY 



is the equation of a plane through the origin perpendicular to cpy, and 

 bisecting all chords parallel to y. But such a plane must be that of 

 a and /3, for 



S{xa -\- y^)cpy = xSacpy -\- yS^gij' = 0. 



Thus we have a system of three planes, each of which bisects all 

 chords parallel to the intersection of the other two. Hence, if three 

 vectors a, ^, y are such that 



Sciq>§ = = S|3qpa, 



Sacpy =^ = Syq)a, 



S^qiy = = S;'qp^, 



the diameters parallel to «, /3, y are conjugate diameters, and the planes, 



Srog)« =: 0, Swqp^ = 0, Scoq)y = 



are conjugate diametral planes. The above equations, which may be 

 written in the form 



cos 



< ^« = 0, cos < '''J = 0, and cos < '''^ = 0, 



give three conditions for determining the directions of «, /3, 7 ; since 

 the direction of q)Q depends only on the direction of q. But three 

 directions involve six arbitrary constants, of which we see that three 

 may be selected arbitrarily. Thus, if one diameter, or one plane, be 

 chosen, the other two can still be taken in an infinity of ways. 



Again y, for instance, bisects all chords through it parallel to the 

 plane of a and ^ ; because, \{ d = aa -\- b^, 



Sj'(jp5 =: aS;'qp« -\- bSycp^ = 0. 



Hence the equation 



^(tny ± xd)rp(^my ± xd) 



= m^Syqiy -\- x'^Sdqid ± 2 rnxSycfd 



= m^Syq}y -\- x^SdqiS =1, 



is satisfied by equal and opposite values of x. 



