442 Transactions . — MisccllancoiLS. 



made to assume all possible positions, any point, P, in the 

 connecting bar will describe a curve called a lemniscoid, 

 of the general shape of an elongated figure of eight. At the 

 point of crossing of the two branches a portion of either is 

 very approximately a straight line, and thus if the rods AB 

 and CD do not turn through too great an angle, P may be 

 attached to a piston- or slidevalve-rod, which is constrained 

 to move in a straight line, wdthout danger of breaking the 

 machinery. 



The arrangement thus suggested would require a greater 

 space for the machinery than is ordinarily available if CD 

 represented the half of the beam and P the point of attach- 

 ment of the piston-rod. By means of an arrangement of 

 parallel rods the motion of P is multiplied, so to speak, in the 

 following manner : — 



For simplicity's sake, suppose that AB and CD are equal, 

 and P the middle point of BD. Imagine CD to be produced to 

 E, making DE equal to CD, and let two other rods, as EQ 

 and QB, equal to BD and DE respectively, be hinged to the 

 others at E and B, and to each other at Q. 



Then elementary geometry shows that throughout the 

 motion DEQB is always a parallelogram, and, since QE is 

 double of DP, and CE double of CD, the points C, P, Q are 

 in a straight line, and CQ is always double of CP. Hence 

 the path described by Q must be similar to that of P on a 

 scale twice as large, and, as P moves approximately in a 

 straight line, so also will Q. 



CE represents half the beam, and Q is the point of attach- 

 ment of the piston-rod, while P serves as a point of attach- 

 ment of a pump- or valve-rod. 



These motions are illustrated by the models shown. The 

 problem of connecting an exact rectilinear motion with a 

 circular one has only been solved in comparatively recent 

 times, the first arrangement of link-work effecting this object 

 having been devised by M. Peaucellier, a French engineer 

 officer, in 1864. Other methods of achieving the same result 

 have since been discovered. 



The geometrical theorem on which M. Peaucellier' s 

 apparatus depends is the following : — 



