228 KANSAS ACADEMY OF SCIENGK. 



q' as axes to rabatte the planes through C, and II into II' without 

 changing the figures situated in these planes. After the motion there 

 is still OR = and || Q'S and SP' = SP', so that the distances PR and 

 PS remain also unchanged. From this it follows that after the mo- 

 tion P' and the rabatted position of P lie on a ray through the re- 

 volved position of C. The laws expressing the relations between the 

 revolved and projected figure are therefore the same as those between 

 the figure in sjjace ( II ) and its projection II'. After the rabatte- 

 ment, fig. 5 assumes the form of fig. 6. Here 1 and 1' are the two cor- 

 responding lines which with s and SC form a pencil of four rays 

 through S. As OP and CQ intersect this pencil there is 

 (CLP'P) = (CMQ'Q). 



CO' CO 

 The value of ( CMQ'Q ) is ^^, = ^^ = k, ray ; i. e., entirely independ- 

 ent of the position of 1, 1', and CP. Thus, drawing any ray through 

 and intersecting s in S', any two points P and P' on this ray of the 

 central projection form a constant ratio with C and S'. For all pos- 

 sible pairs P and P' of corresponding points 



(CSPP') = constant. 

 The different cases of central projection may be classified according 

 to the position of the center of projection and the value of the con- 

 stant k.* 



Laying any Carterian system of coordinates through 0, and desig- 

 nating the coordinates of any pair of corresponding points P and P' 

 by X, y, and x', y', respectively, it is an easy matter to derive from the 

 figure the general form of the relation existing between these coordi- 

 nates : 



, ax '"1 



^ ~dx + ey + f' I 



y (1) 



y ~dx + ey + f' I 

 where - is the constant k of the projection. 



Formulas (1) represent the transformation of a point P into P', 

 called perspective. 



The equation of the line r is dx + ey + f ^ o, and its correspond- 

 ing line r' is infinitely distant. To the line q (x = go, y = oo) corre- 

 sponds the line q' with the equation dx'+ ey' — a = o. The axis s 

 is obtained by putting x':=x, y'=y, and its equation is dx + ey + f 

 — a = o. Comparing the equations of the lines r, s, q', and their dis- 

 tances from C, it is found that 



f — a _ f a 



1 ' (F+e"' v' d-+e'' i/ d'+e* ' 



* Fiedler, loc. cit., p. 95. 



