Section III., 1896. [ 2S ] Trans. R. S. C. 



11. — Mechanism for Describiny Conic Sections. 



By J. J. Guest. 



(Read May 20, 1896.) 



Peaucellier's solution of the mechanical description of a straight line, 

 by means of a mechanism containing turning pairs only, incited a large 

 interest in plane mechanisms of that class, and several such linkages have 

 been designed to describe the conies. These, however, are mostly some- 

 what cumbersome, and can usually describe a small portion only of the 

 curve ; by having recourse to the sliding pair, however, simpler means to 

 the same end can be arrived at. As examples, I propose to describe 

 three mechanisms which will draw the desired curve entirely, or until 

 the bars are fully extended. 



The first mechanism describes hyperbolas by making use of the pro- 

 perty that the differences of the focal distances of any point on the curve 

 is constant. In Fig. 1, ABCD is a crossed four-bar linkage, the opposite 

 links of which are equal. If AC and BB, when produced, meet in P, a 

 symmetrical figure is formed and PC = BP, and hence PA — PB = AC^, 

 which is constant. Hence, if two links PCA, PPB are connected to AC^ 

 and BD by sHding pairs parallel to AC and BD respectively, and to one 

 another by a turning pair at P, then P will, in its movement, trace out 

 relatively to the link AB, an hyperbola whose foci are at A and B. 



Constructively AB should form a bridge over the paper (see Fig. 2),. 

 and the Unk CD should lie below the bx'idge but above the rest of the 

 mechanism. The length of the curve which can be described will then 

 be limited only by the length of the slides. This mechanism has a change 

 jjoini when ("' crosses J. i^ ;. the correct motion at this point can readily 

 be enforced by Eouleaux' method. 



The effect of friction on locking in a turning pair can only be deter- 

 mined when the diameter of the journal is known, but in a sliding pair it 

 occurs whenever the force on the slide makes an angle less than the 

 angle of friction with the normal to the direction of relative motion. 

 This, however, will never occur in the above mechanism if the movement 

 is produced by a string attached to P and pulled in a direction approxim- 

 ately bisecting the angle APB. 



The second mechanism makes use of Mr. Kempe's variation of the 

 Hart cell to describe hyperbolas referred to their asymptotes. In this 

 mechanism, if LSM, 31PK, KQN, and NOL (Fig. 3) be similar triangles 

 described upon the bases i3/, MK, KN, NL of the Hart contr^paral- 

 lelogram, then OSPQ is a parallelogram of constant area. Hence, by 



Sec. III., 1896. 3. 



