[ 304 ] 



XLITI. Proceedings of Learned Societies, 



ROYAL SOCIETY. 



[Continued from p. 232.] 



June 17, 1869. — Lieut. -General Sabine, President, in the Chair. 



THE following communication was read : — 

 " On the Mechanical Description of Curves." By "W. H. L. 

 Russell, F.R.S. 



Let A, B, C be three wheels rolling in one another (fig. 1 ) ; 

 they may of course be supposed to describe simultaneously the angles 

 md, nd, rd, when m, n, and r are constant. 





Fig. 1 



• 







8IB 













Let a, /3, y be three nuts situated on A, B, C respectively, at dis- 

 tances a, b, c from their centres. Then if these nuts work in hori- 

 zontal bars (as exemplified in many sewing-machines), the bars will 

 descend vertically through the spaces a sin md, b sin nd, c sin rO 

 respectively. We may combine all these vertical motions together ; 

 for if vertical rods be attached to the horizontal bars, and a cord fixed 

 at Q pass over the pulleys a v A 2 , a 3 , b v B 2 , b 3 , e l3 C 2 , c 3 , as shown in 

 the figure, the other extremity Q x will describe the space a sin md + b 

 sin nd + C sin rd. By this contrivance we are able to combine any 

 number of vertical descents, so that it is readily seen that a sin 

 (md + ct) + b sin (ra0-f/3)+ &c. may be described mechanically. A 

 machine on the same principle as this had been previously invented 

 by Mr. Bashforth. 



I soon perceived that in order to describe the general equation 

 of the rth order by continued motion, it was necessary to make a 

 wheel revolve through an angle equal to the sum and difference of 

 the angles described in the same time by two given wheels ; to effect 

 this I invented the apparatus shown in fig. 2. 



