PROFESSOR TAIT ON IMPACT. 227 



to be approximately" constant, it is easy to see that the equation of the part of the tracing 

 made during a fall, or during a rise, can be put in the very simple form 



r = A + B0 2 . 



Here the centre of the disc is the pole, and the initial Hne is the particular radius 

 which was vertical when the block was at one of its successive highest positions. This 

 radius separates the rise, from the fall, part of each branch of the curve. A is of course 

 the same for both parts, but B (being directly as the acceleration of the block, and inversely 

 as the square of the angular velocity of the disc) is larger for the rise than for the fall ; 

 because friction aids gravity in the ascent and acts against it in the descent. A number 

 of sets of corresponding values of the polar coordinates were measured on each part of 

 the curve, the angles being taken from an approximately assumed initial line. Three of 

 these sets determined A, B, and the true position of the initial radius ; and the others 

 were found to satisfy (almost exactly) the equation thus formed. This shows that the 

 assumption, of friction nearly constant throughout the whole trace, is sufficiently 

 accurate. B is always positive in the equation, but A is negative or positive according 

 as the block does, or does not, rebound to a height greater than the radius of the datum 

 circle. 



It is not necessary to tabulate here any of the very numerous results of these earlier 

 experiments. While the work was in progress many valuable improvements of the apparatus 

 suggested themselves, and I resolved to repeat the experiments after these had been intro- 

 duced. The whole of these subsequent results are tabulated below. The following were 

 found to be the chief defects of the earlier arrangement, so far at least as they were not 

 absolutely inherent in the whole plan. These have been since remedied ; and results 

 obtained with the improved apparatus have been, from time to time, communicated to 

 the Society. 



1. The use of a pencil is objectionable from many points of view. Serious worry and 

 much loss of time are incurred in consequence of the frequent breaking of the lead, even 

 when every possible precaution seems to have been taken. Then the rapid wearing-down 

 of the point by the cartridge-paper causes the later-traced portions of each diagram 

 (including especially the datum circle, which is of vital importance) to be drawn in broad 

 lines, whose exact point of intersection can be but roughly guessed at. The friction, also, 

 was (mainly on account of the roughness of the paper) so large that the values of B, for 

 the ascending and descending parts of any one branch of the curve, differed from the 

 mean by a large fraction of it, sometimes as much as 20 per cent. This is approximately 

 the ratio which the acceleration due to friction bears to that due to gravity ; so that the 

 friction was, at least occasionally, as much as one pound weight. This, of course, seriously 

 interfered with the accurate measure of the coefficient of restitution. Instead of the board 

 and cartridge-paper I introduced a specially prepared disc of plate-glass, which ran per- 

 fectly true. It was covered uniformly with a thin layer of very fine printers' ink, which 

 was employed wet. For the pencil was substituted a needle-point, so that this part of the 



