﻿288 Prof. Cayley on the Plane Representation 



two coincident points) represents the intersection of the line L 

 with the plane of projection. 



The line through the points (P', P") and (Q", Q") has for 

 its projections the lines P' Q' and P" Q". 



Two lines (L/, L") and (M 7 , M") intersect each other if only 

 the intersections 1/ M' and L" M" are the projections of a point 

 P— that is, if the line through the points L' M' and L" M" 

 passes through 12. And then clearly P is the intersection of the 

 two lines. 



A plane TT is conveniently given by means of its trace © on 

 the plane of projection, and of the projections (P', P") of a point 

 on the plane ; or, say, by means of the trace ©, and of a point P 

 on the plane. 



Suppose, however, that a plane is given by means of a line L 

 and a point P on the plane. The trace © passes through the 

 point of intersection of the line L with the plane of projection — 

 that is, through the point of intersection of the projections L', L". 

 To find another point on the trace, we have only to imagine on 

 the line L a point Q, and, joining this with P, to suppose the line 

 P Q produced to meet the plane of projection. The construction 

 is obvious ; but by way of illustration I give it in full. Through 

 11 draw a line meeting L', L" in Q', Q" respectively (then these 

 are the projections of a point Qon the line L) ; the lines P' Q' 

 and P" Q" are the projections of the line P Q, and the intersec- 

 tion of P' Q' and P" Q" is therefore the required point on the 

 trace ©. 



The line of intersection of two planes passes through the point 

 of intersection of their traces % x , © 2 ; whence, if the planes have 

 in common a point P, the line of intersection is the line joining 

 P with the intersection of the traces © x , (B) 2 . 



In what precedes we have the solution of the following pro- 

 blem : — " Given a point P, and two lines L 1? L 2 , to find a line 

 through P meeting the two lines L x , L 2 ." The required line is 

 in fact the line of intersection of the planes (P, LJ and (P, LJ ; 

 we have seen how to construct the traces S 1 and ® 2 of these 

 planes respectively ; and the required line is the line joining P 

 with the intersection of ® t and © 2 . 



I proceed now to the problem to find the two lines, each of 

 them meeting four given lines, L 1? L 2 , L 3 , L 4 (these being, of 

 course, given by means of their projections (L/ x , L/'J &c). The 

 question is in effect to find on the line L t a point P such that, 

 drawing from it a line to meet L 2 , L 3 , and also a line to meet L 2 , 

 L 4 , these shall be one and the same line. 



Now, considering in the first instance P as an arbitrary point 

 on the line L 15 the. line from P to meet L 2 , L 3 is any line what- 

 ever meeting the lines L 1? L 2 , L 3 : say it is a generating line of 



