Mr. 0. V. Boys on an Integmting-Mackine. 345 



and in illustrating those principles themselves, I think it would 

 be found of very great value. To justify myself for bringing 

 a subject purely mathematical before this Society, I will briefly 

 give a few examples of its use. For simplicity's sake let k= 1. 



If A is moved along the axis of x (that is, y = 0), the cart 

 draws a horizontal line, the ascent is nothing, and the area is 

 nothing. By this means any want of parallelism between 

 the front wheel F and AB can be detected and set right (see 

 Plate VIII. fig. 1). 



If A is moved along a line parallel to the axis of x (that is, 

 y = c), the cart draws the straight line y=cx\ that is, the 

 inclination is constant, showing that area is passed over uni- 

 formly (see Plate VIII. fig. 1). 



If A is moved along an inclined straight line y == ex, the 



cx^ 

 cart draws the parabola y— -^- (see Plate VIII. fig. 2). This 



is the path of a projectile : and the machine proves that it 

 must be so ; for taking abscissas as time, the curve repre- 

 senting the velocity of falling is an inclined straight line, while 

 the space fallen in any time, being measured by the area 

 between the inclined line and the axis of x up to that point, is 

 found by the cart ; and as the horizontal movement is propor- 

 tional to the time, the curve drawn by the cart is the path of 

 a projectile. 



If A is moved along the curve y= —5-, a curve representing 



x 



attraction, the cart draws a rectangular hyperbola, showing 

 that potential varies inversely as distance. As abscissas are 

 distances and ordinates forces, it is plain that the work done by 

 an attracting body in bringing a unit from an infinite dis- 

 tance up to a point (that is, the potential at that point) is 

 measured by the area between the curve, the axis of x, and 

 the ordinate at that point ; and as in finding this area the 

 machine draws a rectangular hyperbola in which, of course, 

 y varies inversely as x, it proves that potential varies inversely 

 as distance (see Plate VIII. fig. 8). 



If A is moved along the logarithmic curve y=-e x , the cart 

 draws an identical curve ; and this it should do, since 



— c=e x (see Plate VIII. fig. 4). Since the pointer A and 



the cart describe identical curves, it is plain that their distance 

 asunder is constant ; if, therefore, these two are connected by 

 a link, and then the machine is started on the axis of x, they 

 will each describe a horizontal line. But this will be an un- 

 stable motion ; for if they depart ever so little from horizontal 

 Phil. Mag. S. 5. Vol. 11. No. 69. May 1881. 2C 



