Sept. 26, 1878] 



NATURE 



573 



hole at G, to the bob, and fasten it there to the cords. 

 Then get a small bit of copper wire and bend it once 

 round the two cords just above the knot, as at r in the 

 figure. This wire ring, and the upright post at the side 

 of the platform, we do not need at present, but they will 

 be used in future experiments with this pendulum. 



Tack on the platform AA a strip of wood I. This 

 serves as a guide, along which we can slide the small 

 board m, on which is tacked a piece of paper. 



Experiment 5, — Fill the funnel with sand, and, while 

 the pendulum is stationary, steadily slide the board under 

 it. The running sand will be laid along L M, Fig. 4, in a 

 straight line. If the board was slid under the sand 

 during exactly two seconds of time, then the length of 

 this line may stand for two seconds, and one-half of it 

 may stand for one second, and so on. Thus, we see how 

 time may be recorded in the length of a line. 



Brush off the heaps of sand at the ends of the line, and 



Fig. 4. 



bring the left-hand end of the sand-line directly under 

 the point of the funnel, when the latter is at rest. Draw 

 the lead bob to one side, to a point which is at right 

 angles to the length of the line, and let it go. It swings 

 to and fro, and leaves a track of sand, a b, which is at 

 right angles to the line L M, Fig. 4, 



Suppose that the pendulum goes from a to b, or from 

 b to a, in one second, and that, while the point of the 

 funnel is just over L, we slide the board so that, in two 

 seconds, the end M of the line L M comes under the point 

 of the funnel. In this case the sand will be strewed by 

 the pendulum to and fro, while the paper moves under it 

 through the distance L M. The result is that the sand 

 appears on the paper in a beautiful curve L c N D M. 

 Half of this curve is on one side of L M, the other half 

 on the opposite side of this line. 



The experimenter may find it difficult to begin moving 

 the paper at the very instant that the mouth of the funnel 



Fig. 5. 



is over L ; but, after several trials, he will succeed in 

 doing this. Also, he need not keep the two sand-lines, 

 L M and a b, on paper during these trials ; he may as 

 well use their traces, made by drawing a sharply-pointed 

 pencil through them on to the paper. 



By having a longer board, or by sliding the board 

 slowly under the pendulum, a trace with many waves in 

 it may be formed, as in Fig. 5. 



As the sand-pendulum swung just like an ordinary 

 pendulum when it made the wavy lines of Figs. 4 and 5, it 

 follows that these lines must be peculiar to the motion of 

 a pendulum, and may serre to distinguish it. If so, this 

 curve must have some sort of connection with the motion 

 of the conical pendulum, described in Experiment 4. 

 This is sOy and this connection will be found out by an 

 attentive study of Fig. 6. 



In this figure we again see a wavy curve, under the 

 same circular figure which we used in explaining how the 



mrf to 



y 



motion of an ordinary pendulum may be obtained from 

 the motion of a conical pendulum. This wavy curve is 

 made directly from measures on the circular figure, and 

 certainly bears a striking resemblance to the wavy trace 

 made by the sand-pendulum in Experiment 5. You v/ill 

 soon see that to prove that these two curves are precisely 

 the same is to prove that thej apparent motion of the 

 conical pendulum is exactly like the motion of the 

 ordinary pendulum. 



The wavy line of Fig. 6 is thus formed : — The dots on 

 A B, as already explained, show the apparent places of 

 the ball on this line, when the ball really is at the points 

 correspondingly numbered on the circumference of the 

 circle. "Without proof, we stated that this apparent 

 motion on the line A B was exactly like the motion of a 

 pendulum. This we must now 

 prove. The line L M is equal to „ 



the circumference of the circle 

 stretched out. It is made thus : — 

 We take in a pair of dividers the 

 distance i to 2, or 2 to 3, &c., 



from the circle, and step this ^\ ~\ — p — ^--t-jfi" 



distance off sixteen times on the 

 line L M ; hence L M equals the 

 length of the circumference of the 

 circle, hi time this length stands 

 for two seconds, for the ball in 

 Experiment 4 took two seconds Aj 

 to go round the circle. This same 

 length, you will also observe, was 

 made in the same time as the sand- 

 line L M was made in Experiment 

 5. In Fig. 6 the length LM, of 

 two seconds, is divided into sixteen 

 parts ; hence each of them equals 

 one-eighth of a second, just as the 

 same lengths in the circle equal 

 eighths of a second. Thus the 

 line L M of Fig. 6, as far as a 

 record of time is concerned, is 

 exactly like the sand-line L M of 

 Experiment 5, and the line A B of 

 Fig. 6, in which the ball appeared 

 to move, is like the line ab oi 

 Fig. 4, along which the sand-pen- 

 dulum swung. 



Now take the lengths from i to 



2, I to 3, I to 4, 1 to 5, and so on, 

 from the line A B of Fig. 6, and 

 place these lengths at right angles 

 to the line L M at the points i, 2, 



3, 4, 5, and so on ; by doing so, ^\ 

 we actually take the distances at 

 which the ball appeared from i 

 (its place of greatest velocity), 

 and transfer them to L M ; there- 

 fore, these distances correspond 

 to the distances from L M, Fig. 4, 

 to which the sand-pendulum had "" 

 swung at the end of the times Fig. 6. 

 marked on L M of Fig. 6. 



Join the ends of all these lines, 2 2', 3 3', 4 4'> &c-; by 

 drawing a curve through them, and we have the wavy 

 line of Fig. 6. 



This curve evidently corresponds to the curve L C N D M 

 of Fig. 4 made by the sand-pendulum ; and it must be 

 evident that, if this curve of Fig. 6 is exactly like the 

 curve traced by the sand-pendulum in Experiment 7, it 

 follows that the apparent motion of the conical pendulum, 

 as seen in the plane in which it revolves, is exactly like 

 the real motion of an ordinary pendulum. 



Experiment 6. — To test this, we make on a piece of 

 paper one of the wavy curves exactly as we made the one 

 in Fig. 6, and we tack this paper on the board L M of 



