i905\ 



2' 



Henderson — A Memoir. 



129 





a 13)68' Cx -f (yS' — y'8)aa Bzv = 

 a'/3) yy'Zty + (y$' — JS)PFAz = 



following- conclusion, which is 



Hence we reach the 

 SchlafiTs theorem: — 



The five lines determined from five co-tractorial lines by 

 choosing the remaining tractor in each sd of four of the latter 

 lines, are the?nselves co-tractorial. 



In the above proof, the complete set of lines was derived 

 from the five co-tractorial lines 1', 3', 4', 5', 6', but it is imma- 

 terial from which five of the primed or unprimed lines we 

 start. Moreover the relation between the sets 1', 3', 4', 5', 6' 

 and 1, 3, 4, 5, 6 is a reversible one — the lines of one set are 

 the tractors of the other set by fours and vice versa. 



§8. Anharmonic Ratios. 



Let us next find the co-ordinates of the points of intersec- 

 tion of the lines 2', 3', 4', 5', 6' with the line 1. Determining 

 these in the usual way and writing down also the co-ordinates 

 of the vertex C of the fundamental tetrahedron ABCD, we 

 tabulate them as follows;— 



p 



IS' 



: 



fl8'(y 



'8 — 



yB')A 



yy 



XaF-a'fi)l> 







p 



13' 







/TS'y- 



- Kfiy' 





rt'D 







P 









1 















P\ 



IS' 









ft 







y 







p 



16/ 









fi 







y 







c 



: 













l 







The anharmonic ratio of the four collinear points P , P , P 



12' 13' IS' 



P is identical with the anharmonic ratio of the four para- 



16/ 



meters 



