﻿of a Solid Figure. 289 



the hyperboloid whose directrices are L t , L 2 , L 3 , or of the hyper- 

 boloid LjLgLg. Hence projecting from any point 12' whatever, 

 the generating lines and directrices are projected into tangents 

 of one and the same conic. We know the projections L^, L/ 2 , 1/ 3 

 of the directrices ; to find two other tangents of the conic, we 

 take two arbitrary positions of P on the line L x , and construct 

 as above the projections M', N' of the lines from these to meet 

 the lines L 2 , L 3 . The conic is then given as the conic touching 

 the five lines, TJ V L/, L' 3 , M', N' : say this is the conic 2'. Simi- 

 larly, instead of 12', considering the point 12", we have the lines 

 U' v L" 2 , L" 3 and the lines M", N", which are the other projec- 

 tions of the lines through the two positions of P; and touching 

 these five lines we have a conic 2". Each tangent T' of 2', com- 

 bined with the corresponding tangent T" of 2", represents a line 

 T meeting L x , L 2 , L 3 ; to establish the correspondence, observe 

 that, inasmuch as the line T meets L x , the intersections of T', L' x 

 and of T", U\ must lie in a line with 12 ; if T' be given, the 

 point T" I/'j is thus uniquely determined, and therefore also T" 

 (since U\ is a tangent of 2") ; and similarly if T" be given, T' is 

 uniquely determined ; the correspondence T', T" is thus, as it 

 should be, a (1, 1) correspondence. 



Considering in like manner the lines which meet L x , L 2 , L 4 , 

 we have touching U v L' 2 , L/ 4 , M', N' a conic 2' ; and touching 

 L" v L" 2 , L" 4 , M", N" a conic 2" ; each tangent f' of %', com- 

 bined with the corresponding tangent T" of 2", represents a line 

 meeting L v L 2 , L 4 , the correspondence being a (1,1) corre- 

 spondence such as in the former case. 



The conies 2^2' both touch L' x , L' a ; hence they have in 

 common two tangents. Say one of these is T'= V, the corre- 

 sponding tangents T" and T" will coincide with each other and 

 be a common tangent of 2", 2" (these conies both touch L" x , L" 2j 

 and have thus in common two tangents). We have thus T' = T', 

 and T"=T" as the projections of a line meeting L x , L 2 , L 3 , L 4 ; 

 and taking the other common tangents of 2', 2' and of 2", 2", 

 we have the projections of the other line meeting L x , L 2 , L 3 , L 4 . 



The whole process is :— Construct M', M" and N', N", each of 

 them the projections of a line through a point P of L x , which 

 meets L 2 , L 3 ; and M', M" and N', N", each of them the projec- 

 tions of a line through a point P of L x , which meets L 2 , L 4 ; we 

 have then the conies 



X 1 , 2" touching L' l5 1' 2 , L' 3 , M', N', and L" x , L" 2 , L" 3 , M", N" 

 respectively, 



f',2" „ L'^V^W, „ L'^L'^L^M'^N" 

 respectively ; 



