Mr. C. V. Boys on an Integrating-Macliine. 343 



its steepness, as measured by the tangent of the inclination, for 

 any value of x may be proportional to the ordinate of the 

 given curve for the same value of x. The ascent then made 

 by the new curve in passing from one ordinate to the other is 

 a measure of the area required. 



On Plate VIII. is a plan and side elevation of a model of 

 the instrument made merely to test the idea : the arrange- 

 ment of the details is not altogether convenient. The frame- 

 work is a kind of T-square carrying a fixed centre B, which 

 moves along the axis of x of the given curve ; a rod passing 

 always through B carries a pointer A, which is constrained to 

 move in the vertical line e e of the T-square ; A then can be 

 made to follow any given curve. The distance of B from the 

 edge e e is constant ; call it k : therefore the inclination of the 

 rod AB is such that its tangent is equal to the ordinate of the 

 given curve divided by k ; that is, the tangent of the inclination 

 is proportional to the ordinate ; therefore, as the instrument is 

 moved over the paper, AB has always the inclination of the 

 required curve. 



The part of the instrument that draws the curve is a three- 

 wheeled cart of lead whose front wheel F is mounted, not as 

 a castor, but like the steering-wheel of a bicycle. When such 

 a cart is moved, the front wheel F can only move in the 

 direction of its own plane, whatever be the position of the 

 cart ; if, therefore, the cart is so moved that F is always in 

 the line e e and at the same time has its plane parallel to the 

 rod A B, then F must necessarily describe the required curve; 

 and if it is made to pass over a sheet of black tracing-paper, 

 the required curve will be drawn. 



The upper end of the T-square is raised above the paper, 

 and forms a bridge, under which the cart travels. There is a 

 longitudinal slot in this bridge, in which lies a horizontal 

 wheel, carried by that part of the cart corresponding to the 

 head of a bicycle ; by this means the horizontal movement 

 communicated to the front wheel of the cart by the bridge is 

 equal to that of the pointer A ; at the same time the cart is 

 free to move vertically. It only remains to describe the 

 mechanism which causes the plane of the front wheel of the 

 cart to be always parallel to A B. For this purpose I make 

 use of the principle of the epicyclic train. If three equal wheels 

 are mounted on an arm, with their 

 centres in a straight line and their 

 edges in contact, any motion may 

 be given to the arm or to the first 

 wheel, a, yet lines on the first and 

 last wheels, a and b, if ever parallel, 



