TBANSACTIONS OF SECTION A. 



539' 



Fig. 2. 



the straight line) may be regarded as the centre of similitude of the circle and. 

 straight line. 



The changed construction is shown in iig. 2, 

 where P'K is parallel to SO. We have 



RQ : RP' = SQ : SO = constant 

 = SR : HP' -SO. 



Fig. 3. 



Thus R is on a conic, focus S, directrix parallel to 

 LP'. The above suggests the following correspond- 

 ence: 



Let S, be two fixed points, XLy a plane per- 

 pendicular to SO (though this restriction may easily 

 be removed). Let P be any point in space, and let OP meet the plane in X ;: 

 then a parallel XP' to SO meets SP in a 

 point P' corresponding to P. 



If P move over a plane, P' lies on 

 another (intersecting the first on the plane 

 XLy). Therefore if P describe a straight 

 line, P' describes another. 



If P lie on a surface, P' lies on another *S' 

 of the same degree. 



The following theorem determines the 

 analysis, viz. : — 



|?.°t = Co,PSO, 



or, if SL be axis of .r ; SO = a ; OL = b; _ + _ = 1, 



Hence if .v, y, z be co-ordinates of P ; x ,y', z', of P' (S being origin), 



X _y _s 

 x' y' z' 

 Thus, the plane 



is transformed into the plane 



X — a_ a 

 '~6~~x^^ 



(!)• 



Ix + my + nz = ka 

 Ix' + my' + nz' = k(x' — h), 



and so on. 



If we take SO = OL, the correspondence becomes involutional in character, i.e.f 

 if P' correspond to P, then P will correspond to P'. 



In that case the point S may be dropped, 

 and we have the following construction : — 



is a fixed point, LXX' a fixed plane; 

 P is any point in space, PX' perpendicular to 

 plane LXX' ; OX' meets a parallel through X 

 to PX' in P'. Then P' corresponds to P. 



Taking O as origin, and co-ordinates as 



(2) 



y = mx + nd becomes «/ = r?x + md ; 



and so on. 



Since PP' passes through a fixed point S in OL, we see that perspective projec-- 



