40 Ellery W. Davis 



involution. Similarly tan 6" is the symbol of the involution 

 — tan 0", o, tan 0" , co cut out on the perpendicular to the ter- 

 minal line of 9' through its tip P' . 



These statements can all be modified to fit the generalized pro- 

 jected coordinates. In place of perpendicular lines we have lines 

 harmonic with regard to the circular rays through their intersec- 

 tion, say gz^a-M-perpendicular. 1 In place of parallel lines we have 

 lines meeting on s } say ^Maw-parallel. Instead of equal line- 

 segments, vectors and angles we have line-segments, vectors and 

 angles whose dimensions, as expressed in the projected coordi- 

 nates, come out equal. Thus fig. 31 goes over into something 

 like fig. 34, where OX x is the terminal line of 6' and OY \ is quasi- 

 perpendicular thereto. In fact, all lines meeting in Y r are quasi- 

 perpendicular to each line through X lt just as all meeting in Y 

 are quasi-perpendicular to each through X. 



Fig. 35. 



Others of our figures can be similarly projected. In particular, 

 it will be interesting to consider the various constructions for a 

 third element given two elements of a line, figs. 9, 10, n, 12 

 and 13. 



1 Salmon, Higher Plane Curves. 



40 



