196 Mr. C. V. Boys on Apparatus 



must, from the mechanical construction, be the same in the 

 disk and in the ring); and the other, a lateral motion, propor- 

 tional to p c, with which the ring in no way interferes. In 

 a similar way, the motion of S x may be resolved into two — 

 a downward motion proportional to $ip, and a lateral motion 

 proportional to p e. The downward motion must be common 

 to the disk and ring, while the lateral motion is free. It is clear 

 then, if the rings move with speeds in the ratio S/> : S x />, that 

 the centre c will have no up or down tendency ; and if the rings 

 are not moving with speeds in the ratio Sp : Si jp, that the centre 

 o will move up or down the slope with a vertical speed propor- 

 tional to the distance of the point p from a point which does 

 divide S Si in the ratio of the speeds of the smoke-rings. If 

 6 is the inclination of the dotted line, then c will move along 

 this line with a speed equal to cosec 6 times its vertical 

 speed, and p will travel along S Si with a speed equal to cot 

 times the vertical speed of c . Should one disk ever stop and 

 change the direction of its motion, then c must move along 

 the slope till it is immediately under the ring, and move 

 beyond till it arrives at such a position c x that ^>p x : Sij!?i is 

 the ratio of the speeds. 



Let the two smoke-rings be turned by two integrating 

 machines as already described, then either or both may be 

 going at a variable speed. There must at every moment be 

 some point x in the line S S x such that S x : S x x is the ratio of 

 the speeds. This point will sometimes coincide with P, at 

 which times C will be stationary; it will generally, however, be 

 distant more or less from P, in which case P will pursue it with 

 a speed proportional to its distance from it. If the arm which 

 carries C carries also a pencil bearing against a uniformly 

 travelling band of paper, then the curved line drawn will 

 show what has been the efficiency, horse-power per hour, or 

 whatever it was set to find during any period of time. The 

 travelling band would have to be ruled across with lines show- 

 ing time, and longitudinally with lines showing ratios, the 

 scale being of the kind shown in fig. 3. 



If the slope or the direction of motion had been in the op- 

 posite direction to that shown, then p, instead of approaching 

 its places, would have fled from it with a speed proportional to 

 its distance from it. 



I think it possible that the logarithmic divider might be 

 applied to solve some difficult problems ; for while in action 

 the inclination of the path of the centre c to the line S S 1; or 

 the position of either ring on their common axis, may be 

 changed in any way without interfering with the freedom of 

 the motion of c. 



