1869.] 



Description of Curves. 



73 



through the spaces a sin ?nd, b sin nO, c sin rd 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 lt A 2 , a 3 

 b v B 2 , b 3 > e lt C 2 , c 3 , as shown in the figure, the other extremity Q t will de- 

 scribe the space a sin md + b sin n6 + 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 + a) + 6 sin (w0-f/3)4- &c. may be described mechani- 

 cally. 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 appa- 

 ratus shown in fig. 2. 



Fig. 2. 



In fig. 2 let A be a vertical wheel working truly in a horizontal rack 

 which propels the horizontal frame a, /3, y, I. On this frame stand the 

 wheels B and D parallel to the plane of the paper. The wheel C, supposed 

 perpendicular to the plane of the paper, works by teeth in the wheels B 

 and D, and the four wheels A, B, C, D are precisely equal. 



To the centre of C is attached a square axis, which passes through the 

 centre of the wheel E, so that the wheel E in revolving may, without 

 changing its plane, communicate motion to C as the frame moves forward. 

 Two horizontal racks, R 2 , R 3 , parallel to the plane of the paper, are urged by 

 the wheels B and D ; and these, again, work in the fixed wheels F and G, equal 

 to A, B, C, D in all respects. Then if the wheel A describe in a given 

 time the angle 0, and the wheel E in the same time the angle <p, the wheels 

 F and G will revolve respectively in the same time through the angles 

 0+0 and 0 — 0. 



We shall call the wheel A an abscissa wheel, the wheel E, an ordinate 

 wheel, for reasons which will appear directly, also F an addition wheel, 

 and G a subtraction wheel. 



