Mr. A. Day on the Rotation of the Pendulum. 23 



our index-line will not rotate once during the revolution of its 

 disc over the entire surface of the rack on the conical surface ; 

 and the more acute the cone is the smaller the amount of rota- 

 tion. This is not only analogous to the case of the pendulum, 

 but it is exactly the same in amount, when the plane sectional 

 angle of the cone is altered proportionally to the latitude of the 

 place where the pendulum is set swinging. 



Great confusion has arisen in many minds with reference to 

 this problem from not separating two things, a free and a con- 

 strained and conditioned motion, and not distinguishing what is 

 real from what is apparent. Thus if a cone be set revolving on 

 its axis, a fixed straight line on the surface of the cone will, after 

 one entire revolution, be in the same spot, and we say of it that 

 it has not rotated round any point in itself; but yet if a plane 

 be conceived to pass through the line in its first position and the 

 axis of the cone, and the projection of the line after a quarter 

 revolution falling perpendicular on this plane be drawn, the two 

 lines will cut one another at an angle equal to half the plane 

 sectional angle of the apex of the cone, and in half a revolution 

 this angle will be doubled. The partial rotation however in the 

 first half of its course, which reaches its maximum in half a revo- 

 lution of the cone, is in fact retrograde during the second half, 

 and at the completion of the whole period all things are as they 

 were. The whole of this is a constrained motion, during which 

 every point in the line moves in a circle, but every point with a 

 velocity different from that of its neighbour and in a dijfferent 

 circle. This is totally difi'erent from the case of a line occupying 

 at each infinitesimally-consecutive instant a position parallel to 

 itself, and thus having all its points moving uniformly in equal 

 circles, or constrained to keep to the surface of the revolving 

 cone, so as to cross always a given point on it, or, in other words^j 

 moving at the same rate as the cone. It has been sufficiently 

 and irrefragably demonstrated, that while the cone continues to 

 revolve in the same direction, the line so circumstanced would 

 constantly rotate in one direction round that point which it 

 always crosses ; and that though this result is obtained by the 

 tendency of the line to keep parallel to itself, coupled with the 

 constraint applied to it, it will not, after a complete revolution 

 of the cone on its axis, bring the line into parallelism with its 

 first position, but only after the lapse of a longer period. Hence 

 the rotation is real, not merely apparent', and were there no 

 apparent rotation, there must have been a real retrograde force 

 of rotation applied to the plane of oscillation, so as to have obli- 

 terated all the advance we actually see in practice. I have met 

 with other gentlemen of mathematical education and reputation, 

 who have persisted in calling this rotation an apparent one. 



