26 ROYAL SOCIETY OF CANADA 



fixing and making Q sliile in a straight line passing through 0, tlie 

 point S is forced to describe a straight line through 0^; and P to describe 

 -an hyperbola, of which the paths of S and Q are as^ymptotes. 



Constructively the fixed link containing and the slide should lie in 

 the lowest plane, the lamina LON above that, then the pair LSM i\\\à 

 KQX, and finally MPK. The change points of this mechanism occur 

 when OL, ZM or OX, NK becomes a straight line, after which the 

 mechanism alters its character. 



The third mechanism produces conies by inversion from curves of 

 the form /• = n -\- l> Cos ^. these curves being formed by the double slider 

 crank (TT, SS) chain, the liidv TT being fixed. The sliding pairs are 

 préferablj", though not essentially, at right angles, and the motion of the 

 link >S'*S is kinemetically equivalent to that of a circle rolling internally 

 upon another of half its diameter. For let ,i and B (Fig. 4) be the turn- 

 ing pairs, and C the intersection of lines through A and B parallel to the 

 sliding pairs. Then C"''= OA = OB, and (' describes a circle with angu- 

 lar velocity q'> (where ^ = angle COB), while the link SS revolves with 

 xmgular velocity ê (where d = angle CAB), and we see that (j) = 'IH. 



If .r, y, z be the angular velocities of the first wheel, the revolving- 

 link, and the last wheel in an epicyctic train, we have z — y -\- c (./■ — ?/), 

 and putting x = 0\ y = (J) and z — \(f>, this gives e = 1/2, and ht'uee 

 the motion of *S'*S' is equivalent to that of an internal wheel rolling u])on 

 a fixed wheel of half its diameter, and the motion might be produced in 

 thi.s manner. 



By symmetry it is evident that the paths described by points on l^S, 

 equally distant from C, are the same curves ; and the equation to any of 

 them can therefore be obtained by writing down the equation to the 

 locus of P, a point in AC. Let CP = a and AB = h^ then the locus is 



r = a-\- b Cos 6, which inverts into -. = r (1 + '' Cos H) where e = h/ti = 



AB/CP, and therefore represents any type of conic accoi-ding to the 

 distance of P from C. 



The arrangement is as shown in Fig, 5, the first elements of the turn- 

 ing pairs being fixed to a bridge spanning the paper and the links con- 

 taining the first elements of the slides paired to them. The link >S',S' 

 con.sists of a plate with the two slides cut in it and pierced into holes 

 Q, P, etc.. at various distances from C ; Q, being such that C(^) = AB. 

 The inverting linkage is fixed to the paper at the origin 0, of the curve 

 rz=a-\-b Cos 0, which, by considering the case as represented by the 

 rolling circles, is readily seen to be the position of Q, when CQ is perpen- 

 dicular to AB and C and Q are on opposite sides of AJi. 



YoY inversion the Hart cell has the advantage that the radius of 

 inver-sion can be easily altered, thus producing conies of given eccen- 

 tricity and of various sizes. It also has the great constructive advantag(i 



