for calculating Efficiency. 195 



pointer may be made to trace a diagram on a travelling band 

 of paper. I shall describe one logarithmic, and three har- 

 monic dividers. 



Logarithmic Divider. 

 For this a pair of wheels incapable of steering are required — 

 that is, wheels which when turned while their edges are in 

 contact with a surface are compelled to move forward, but 

 which at the same time are perfectly free to move laterally. 

 Disks with smoothly milled edges very imperfectly fulfil these 

 conditions; but an outside smoke-ring, such as is described in 

 the smoke-ring integrator on p. 79, Phil. Mag. Feb. 1882, should 

 answer this purpose well. Let two such smoke-rings be mounted 

 on a common axis, but so that each may revolve independently 

 of the other ; let there be a disk so supported that its plane 

 is parallel to the common axis, and so that its own axis would 

 if continued meet the other at a point midway between the 

 two smoke-rings ; moreover, let the axis of the disk be carried 

 by an arm in such a manner that it is capable of moving in a 

 direction parallel to the plane of the disk, but inclined at a 

 small angle to the common axis of the smoke-rings. Fig. 1 

 shows the arrangement in its central position : A A is the 

 common axis, S Sj are the smoke-rings, and D the disk ; the 

 dotted line shows the line of travel of the axis of the disk. If 

 while the disk is in its central position the two rings S Si are 

 caused to revolve at equal speeds in opposite directions, then 

 the disk merely turns, and there is no further result. Let the 

 direction of motion be that shown by the arrow. 



Now let the ring Sx begin to revolve faster than S, then the 

 centre of the disk D would, if free, begin to move downwards 

 with a speed equal to half the excess; but it is incapable of 

 moving vertically downwards; yet. it may move down the slope 

 indicated by the dotted line. Now, as the rings S S x in no 

 way interfere with the lateral movement of the disk, its centre 

 will move down the slope till it reaches such a point that the 

 ratio of the distances of the point from the two rings is equal 

 to the ratio of the speeds. It will be necessary to show that 

 what is true when the produced axis of D intersects A A, is 

 equally true when it has so far moved up or down the slope 

 as to cause a considerable displacement. Let the centre have 

 moved down to c, fig. 2, and let S S x be the points on the 

 disk touched by the two smoke-rings, and let cp be the per- 

 pendicular on S S x from c. Now, if S S x are turning with 

 speeds proportional to the lengths S_p and Sip, then the 

 centre will have no tendency to move up or down. The 

 motion of S about c as a centre may be resolved into two: — 

 one, an upward motion, proportional to Sp (and this motion 



