the Projective and the Metrical Scales. 429 



where A 3 = Cos X, A 1 — — Sin X. Therefore, the cross (AY) 

 (BT) is determined by 



^ = ftnX=2i-, ^=2 Tan I. . . . (T) 



Next, we have to find the cross I) o£ the straight lines 



x Tan h + y Tan \ = z Tan h. Tan X, . . (BY) 

 and 



^(l-TanX)Tan h- 2;yTanX=^TanXTan A(TanX-l). (AC) 

 These equations give 



D 1 _ TanX(l + TanX) D 2 2 Tan 6(1 -Tan X) 

 D 3 ~~ 3-TanX ' D 3 ~" 3-TanX * ^ 



Finally, the join o£ D with T, 



I x y z\ 



: T x T 2 T 3 ! =0, 



[ Di D 2 Da ! 



gives, with y = 0, for the abscissa rj = MP of the required, 

 point P, 



Substitute here the values given under (T) and (D) ; then 

 the common factor 2 Tan A will divide out, and the result 

 will be 



/ l- TanX \ 2 Tan 2 X 



V '\ 3-TanX; 3-TanX' . 



independent of //, as was to be expected, or 



Tan 77 = Tan 2 X, 



identical with our previous result. Similarly, for the 

 abscissa 7/ of the other characteristic point, the reader will 

 find Tan rf = — Tan 2 X. 



Thus the result, first obtained by a much simpler method, 

 is fully corroborated. 



Byerson Laboratory, Chicago, 

 July 1921. 



Tan 



