78 Mr. C. V. Boys on Apparatus for the 



machines, comprising Sang's planimeter, Clerk Maxwell's sphere 

 machine, and Sir James Thomson's disk sphere and cylinder 

 integrator (Ashton and Storey's steam-power meter also 

 comes under this head); 2nd, sine or cosine machines, compri- 

 sing Amsler's planimeter and mechanical integrator, and the 

 various wind integrators; 3rd, tangent machines, which, so 

 far as I am aware, are represented only by the cart machine 

 already referred to and those that are the subject of this paper. 

 This class of machines depends on the formula for integration, 

 which, in its geometrical application, finds a curve of which 



the steepness or the tangent of the inclination (i. e. ~) is 



equal to the ordinate of the given curve or to the given func- 

 tion. In my former machine a pointer is made to follow a 

 curve, and by so doing causes a rod to be inclined in such a 

 manner that its tangent is equal to the ordinate. There is 

 also a three-wheeled cart ; and the plane of its steering-wheel 

 is by simple mechanism kept parallel to this rod ; moreover 

 the horizontal component of the cart's motion is equal to that 

 of the pointer. Under these conditions the vertical compo- 

 nent of the motion of the cart (or, shortly, its ascent) deter- 

 mines the integral. 



I will now show how this principle is applied in a series of 

 machines in which the integral is determined by rotation and 

 not by linear motion, and in which, therefore, the integral may 

 grow indefinitely. At first let us suppose that the cart in my 

 first machine is incapable of vertical motion ; then it, in its 

 attempt to move up or down, will push the paper in the oppo- 

 site direction. If now the paper is wound on a cylinder with 

 its axis beneath the path described by the front wheel of the 

 cart, and if the hind wheels are supported by some other means, 

 then the cylinder will rotate ; and the rate of its rotation will 

 be proportional to the ordinate of the given curve, and the 

 amount of its rotation will be the integral required. Now it 

 will at once appear that the cart and the parallel motion are 

 not wanted, and that the inclination of what was the front 

 wheel of the cart, and what may now be called the tangent- 

 wheel, may be determined mechanically by the same method 

 that was adopted to give inclination to the rod. Also if, 

 instead of moving the tangent-wheel along the surface of the 

 cylinder, the cylinder be moved longitudinally under the tan- 

 gent-wheel while its inclination is determined by suitable 

 means, then, as before, the rotation of the cylinder is a mea- 

 sure of the integral. 



As the cylinder must necessarily have a finite length, it 

 cannot be caused to move continuously in one direction under 



