MISCELLANEOUS PAPERS. 227 



with II' is the required point P'. In a similar manner the projection 

 r of a line 1 in II is obtained as the line of intersection of the plane 

 passing through C and 1 with II'. From this construction the follow- 

 ing fundamental laws are immediately clear, and are in fact a mere 

 restatement of the laws of homologous triangles previously considered. 



To every point of II corresponds one and only one point of II', and 

 conversely, and both points are on the same ray through C. 



To every straight line of II corresponds a straight line of II', and 

 conversely ; and both lines meet at the same point of s ( holds for any 

 line). 



To the infinite line q of II corresponds a line q' of II', which is 

 parallel to s. Conversely, to the infinite line r' of II' corresponds a line 

 r parallel to s. 



The plane II is usually determined by its trace s in II' and either 

 of the lines r and q'. If a straight line 1 in II is given, intersecting 

 the trace s in S', the corresponding line 1' is obtained by drawing a 

 line through C parallel to 1 and marking its point of intersection Q' 

 with q'. It is evident that Q' is the projection of the infinitive point 

 of 1, and the projection of 1 consequently passes through S' and Q'. 

 Another way is to produce 1 till it intersects r in R and to join C with 

 R. The line through S parallel to CR is the required projection 1' 

 of 1. From the figure, it is seen that CRSQ' is a parallelogram and 

 that PS : PR = P'S : CR. 



The planes through C parallel to II and II' form a space of a paral- 

 lelepipedon. Keeping II' fixed it is possible by rotations about s and 



