Royal Society, 305 



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

 rack R x , which propels the horizontal frame a, /3, y, I. On this 



Fig. 2. 



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 0, the wheels F and G will revolve 

 respectively in the same time through the angles + and — 0. 



We shall call the wheel A an abscissa wheel, the wheel E an or- 

 dinate wheel, for reasons which will appear directly, also F an ad- 

 dition wheel, and G a subtraction wheel. 



Let x=a sin 0, y=a sin 0, then the general equation of the rth 

 order may be written 



u sin (md + nty) + a sin (m 6— ri(f>) + a" sin (m"d + ri'<p) + ... —a sinB. 



Let a number of machines like the foregoing be placed side by side 

 with their ordinate wheels rolling in one another, and their abscissa 

 wheels duly connected. Let one abscissa wheel describe an angle 

 md, and the corresponding ordinate wheel the angle w0, then a nut 

 placed on the corresponding addition wheel, at a distance a from 

 its centre, will cause a horizontal bar to descend vertically through 

 a space a sin (md + ncp). In the same way a nut properly placed 

 on the subtraction wheel will cause a horizontal bar to descend ver- 

 tically through a space d sin (md—n<p). By means of the adjacent 

 machines we may in like manner cause bars to descend through 

 the vertical spaces, a" sin (m'0 + n'<}>), a" sin (m'0— w f 0), &c. Now 

 let motion be communicated to the ordinate wheels, and let all the 



Phil. Mag. S.4. Vol. 39. No. 261. April 1870. X 



