1879.] Mr Hart, On two models of parallel motions. 317 



infinitely great — in otlier words, P moves in a straight line perpen- 

 dicular to the line joining the fixed points 0, 0'. 



It may be mentioned that the fourth point Q describes a uni- 

 cursal circular cubic (having the above line for its asymptote) 

 which becomes the cissoid when fju^ = 4r^, r being the radius of the 

 circle described by P' (see Messenger, vol. IV. page 117). 



The second parallel motion was first given in the postscript of 

 a paper which I read before the London Mathematical Society (see 

 Proceedings, vol. Vlil. page 288) ; it differs entirely from that 

 given above, and depends on the simple property of a straight 

 line, viz. that it is the locus of a point which moves, so that the 

 difference between the squares of its distances from two fixed 

 points is constant. 



Let a pentagon PEBCF (fig. 2) be constructed of five links, 

 (connected only at the angular points,) whose lengths satisfy the 

 single condition 



PE.EB = PF.FG, 



Fig. 2. 



BC being arbitrary. Let the links be placed so that the angles at 

 E and i^^are equal, as in the figure. Take points A,D in EB, EC, 

 such that 



PE : EA :: CF : FP, 

 and PF : FD :: BE : EP; 



then it is easily seen that, since the triangles PEA, CFP are 

 similar, as are also the triangles PFD, BEP, A D and BG subtend 

 equal angles at P, and hence that PAD and PBC are two similar 

 triangles, and 



