218 UNIVERSITY OF COLORADO STUDIES 



This linkage may be used to solve mechanically two interesting 

 cases of collineations in a plane. If by two Peaucellier Inversors 

 the points A and B are forced to describe the same straight line s, 

 in which an arbitrary point is taken as the origin of a Carterian co- 

 ordinate-system, and s itself is assumed as the cc-axis, we have, since 



AB 



=7)1 (constant), 



.a.ir 



x' =x-\-my, 



(a?, y) and (a?', y') designate the coordinates of P and P' respect- 

 ively. This coUineation, whose group-property can easily be verified 

 by the linkage, is called an elation, according to Lie,^'^ and leaves 

 areas invariant. The point of the line-element is infinitely distant 

 in this case. 



Instead of attaching the rhomb BHP'G as shown in the figure, 

 it may be attached in the opposite direction so that the point P' will 

 now be at P''. Determining the coordinates of P and P" as be- 

 fore, the point P' ' corresponds to P in an oblique axial symmetry 

 with respect to s: 



x' ' z=x-\-my, 



y"=-y- 



Various combinations of the linkages described above, with pan- 

 tographs, Koenig's perspectivograph,^'^) etc., make it possible to realize 

 in a practical form all collineations by linkages. ^^^ The possibility 

 of this is proved by Koenig's general proposition. 



January, 1903. 



(1) Vorlesungen uber Continuierliche Oruppen, p. 262, Teubner, Leipzig:. 



(2) Comptea Rendus, Vol. CXXXI, No. 26, p. 1179. 



{?) This problem has been studied in detail by Hermann Emch, M. S., in his master's thesis 

 — University of Colorado. 



