oltf Mr. C. V. Boys on an Integrating-MacMne. 



motion, they will turn aside faster and faster, the cart pulling 

 the pointer and the pointer directing the cart, and thus origi- 

 nate the logarithmic curve. 



If A is moved along a wave-line symmetrically placed with 

 respect to the axis of &, the cart draws another wave-line a 

 quarter of a wave-length behind the first in point of time. If 

 the first line represents the varying strength of an induced 

 electrical current, the second shows the nature of the primary 

 that would give rise to such a current (see Plate VIII. 

 fig. 5). 



Fig. 6 shows the application of the machine to the deter- 

 mination of the area of a closed curve. 



The rules for finding maxima and minima and points of 

 inflexion are rendered obvious by manipulating the machine. 

 By no means can the cart be made to trace a maximum or a 

 minimum unless the pointer A cross the axis of x ; nor can 

 it pass a point of inflexion unless A pass a maximum or a 

 minimum. 



An indefinite integral requires the addition of a constant ; 

 but on integrating between limits this constant goes out. 

 This is illustrated by the fact that the cart may be started on 

 any level on the board, but the ascent made is the same. 



Not only does the machine integrate y dx, but if the plane 

 of the front wheel of the cart is set at right angles instead of 



dx 



parallel to AB, then the cart finds the integral of — , and thus 



solves problems such, for instance, as the time occupied by a 

 body in moving along a path when the law of the velocity at 

 different parts is known. This is evidently true ; for if a line 

 be drawn perpendicularly to AB through B, it will cut e e at 



a point distant from the axis of x by an amount equal to — , 



and therefore its inclination is such that its tangent is equal to 



— As the cart travels down instead of up for positive values 



of y, its descent instead of its ascent must be taken. 



Some modifications in the instrument would enable it to 

 integrate y 2 dx or y z dx ; it could also be made to integrate 

 the product of two or more functions. I do not intend to go 

 into details with regard to these extensions of the machine, 

 but merely to explain the principle that would be employed. 

 As before, let k=l. To integrate y 2 dx the rod AB would 

 be replaced by a T, as shown in fig. 3. The head of this 

 would obviously cut the axis of x in advance of the edge e e 

 by an amount equal to y 2 . Let a rod pass through this 

 point of intersection and through a point on e e distant from the 



