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



of time, while the distances between them measure the ave- 

 rage rate of growth over each interval. 



I have at present supposed that the integrating surface is 

 cylindrical; but other surfaces of revolution maybe employed 

 for particular purposes. As the rotation of the cylinder de- 

 pends on the linear motion of its surface, it is clear that its 

 rotation must be inversely proportional to its diameter. If, 

 therefore, instead of a cylinder any other surface of revolution 

 is taken, its rate of rotation will depend not only on the inclina- 

 tion of the tangent- wheel, but also on the radius of contact. The 

 simplest case is that of a disk with the tangent-wheel mounted 

 so as to be capable of radial movement. Then, if the tangent- 

 wheel moves in the direction of its own plane, it will simply 

 describe on the disk a radial line, and there will be no rotation; 

 but if it is inclined at any given angle the disk will rotate, and 



the rate of its rotation will be proportional to -, and its whole 



rotation will be j - dr, which is log r. Now the tangent- wheel, 



in its movement outwards, describes on the surface of the disk 

 a spiral which everywhere cuts the radii at the same angle ; 

 therefore in such a spiral the angles are the logarithms of the 

 radii ; i. e. it is the logarithmic spiral If the inclination of 

 the tangent-wheel is made to depend on some function, then 



such double-disk machine would integrate — — d.v, in which 



c is the radius of contact when x = 0. 



If the axis of the tangent-wheel is made to pass through a 

 fixed point over the disk removed from its line of travel by a 

 right angle, then the tangent of its inclination to the direction 

 of its motion is proportional to the radius of contact: but, 

 other things being equal, the rotation of the disk is inversely 

 as the radius of contact ; therefore the amount of rotation of 

 the disk for a given movement of the tangent-wheel is inde- 

 pendent of the radius of contact, and the curve traced out on 

 the disk is the spiral of Archimedes. But if, instead of passing 

 over a disk, the tangent-wheel similarly mounted is made to 

 pass along the surface of a cylinder, then the speed of rotation 

 of the cylinder will be proportional to the distance of the disk 

 from its neutral position, and its whole rotation will be ^cxdx 



or 2# 2 , and the curve described on the cylinder will be a 



parabola. This arrangement of the disk and cylinder may be 

 used, as described on page 84, in a polar planimeter to illus- 

 trate the formula Wrdrdd. 



