•428 



Dr. L. Silberstein on the Relation between 



distances from the axes MY, MB, and from the origin, 

 respectively, the Weierstrass coordinates of N are (with 

 22=1) 



#=Sina, y = $inb, # = Cosr, 



satisfying the condition x 2 + y 2 — z 1 — — 1. All that is 

 required for our purpose is to remember that the equation of 

 a straight line passing through two points w u y ly Z\ and 

 •®2> V^ z 2 1S 



X 



y 



z 



x x 



yi 



z \ 



x 2 



y% 



z 2 



= 0, 



(a) 



and that the angle e contained between any two straight lines 



1, etc.) 



,2_ 



ax + by + cz = 0, a'a:-\-b / y + c'z = (where a 2 + b' 

 is given by 



cos e — aa' + bb' — cc (b) 



It will be convenient to denote the Weierstrass coordinates 

 x, ?/, z of any point N of the plane by JV l5 iVg, A 7 3 respectively. 



Since the position of the required point-pair P, P' is 

 independent of the choice of the point Y, through which 

 the two parallels are drawn, we may take Y on the ordinate 

 axis, in a distance h above the origin M. Thus 



r 1= o, r 2 =sin/i, r 3 =Cos7i, 



The auxiliary line through the end-point B of the segment 

 being arbitrary let us make it perpendicular to AB. In 

 order to find the point P we have to draw through I^the 

 right-hand parallel YL (fig. 7), i. e. the join of Y with the 

 point x, y, c = co , 0, co . Its equation is, by (a), 



tfY 2 +#Y 3 -*Y 2 =0, 



or x + y Cot h = z. ..... (YL) 



The equation of the perpendicular BT is, by (a) and (b), 



x=zTnn\ (BT) 



Thus the ratios of the coordinates of the cross C = (YL) 

 (BT) are 



5i 



= Tan\, 



C 2 



c, 



(l-TanX)TanA. 



The equation of the join A Y is 



xY 2 k 3 +^/Y s A 1 = zA 1 Y, 



(C) 



(AY) 



