142 Mechanical Integration of the Product of two Functions. 



then the area of the figure enclosed by these two curves and 

 the straight lines # = a 1? = a 2 is 



= 2rV(6>)f(6>)d6>. 



J Oil 



Hence all that is necessary for finding the value of the 

 integral is to measure the above area by means of a planimeter. 

 At first sight it might appear as if, in the practical application 

 of the above method, to analyze the arbitrary curve traced by 

 a piece of self-recording physical apparatus into its harmonic 

 constituents, the construction of the curves r = <^(^)±^(^) 

 with any accuracy would be very tedious ; but if it is remem- 

 bered that in all cases the curves actually traced are traced 

 with Cartesian coordinates, it will be seen that to pass from 

 the curves y — $(x), y =-^(x) to the above sum and difference- 

 curves is about as easy as to pass to the curves r = <£(#), 

 r=^(0). 



Wind the two curves y = <£(«#), y = yfr(x) on a circular 

 cylinder so that the axis of x is at right angles to the gene- 

 rating lines, and arrange two pointers for following the curves 

 so that they are free to move parallel to the generating lines 

 and therefore to the axis of y ; attach them to the ends of a 

 string which passes over two fixed pulleys at the level of the 

 top of the cylinder and round a movable pulley, which, as the 

 pointers move, the cylinder remaining at rest, describes a 

 diameter of the top of the cylinder : to enable it to move 

 both backwards and forwards it must be provided with a 

 counterpoise or spring. If the length of the string is so ad- 

 justed that, when the pointers are on the axis of .r, the movable 

 pulley is at the centre of the top of the cylinder, then the dis- 

 tance r of the pulley from the centre will always be the mean 

 of the distances <f>(x) and \jr(x) of the pointers from the axis 

 of x. As the cylinder is caused to rotate round its axis and 

 the pointers are kept on the two curves, a pencil attached 

 to the movable pulley will trace on a sheet of paper on the 

 top of the cylinder the curve r = ^{$(ad) + 'fy(a6)} , where a 

 is the radius of the cylinder. 



By drawing the reflection of the curve yfr(x) with respect to 

 the axis of x and using it with §{x) in the above manner, the 

 curve r = ^{ 4> {ad) —yjr(ad)} can be obtained; or by replacing 

 the fixed pulleys with an obvious arrangement of strings and 

 axles and using the two original curves, the difference-curve 



