PROFESSOR TAIT ON IMPACT, II. 383 



the columns headed N, that the new series of results is at least as trustworthy as the old 

 one. But I was not satisfied with the numbers in the columns A l5 A 2 ; nor, of course, 

 with those in e, which are their respective ratios. These data are derived from the very 

 difficult and uncertain process of drawing tangents at the extremities of portions of 

 curves. I therefore calculated (to two places only) the values of the square-root of the 

 quotient of each pair of successive numbers in the column H. If there were no friction, 

 the results thus obtained should be the successive values of the coefficient of restitution. 

 And, even taking friction into account, if we suppose the acceleration it produces to be 

 ?7i-fold that of gravity (m being, as shown in the first part of the paper, nearly constant 

 and somewhere about 0'03) the values in the table so formed should be those of 



-J\ 



= e(l— m) nearly. 



+ m v J 



This (though at a first glance it might not be suspected) is the result to which we should 

 be led by calculating from the equations of the various parts of the trace the tangents of 

 the inclination of the curve to the radius-vector at the points where it meets the datum 

 circle. For 



cie\ R 



/ , 



tan = fr- 



dr/ n ~2JB(R-A)' 

 so that 



_tan_0 J _ / B 2 (R-A 2 ) _ / B 2 H 2 

 6 ~ tati 2 ~ V B X (R - Aj) ~ V B^j ' 



Unfortunately, it is in general difficult to get a trustworthy value of B for the (first) 

 incomplete branch of the curve. But, by various modes of calculation and measurement, 

 I have made sure that the friction is practically the same whatever be the mass of the 

 block, so that its effects are the less sensible the greater is that mass. The numbers 

 thus obtained fluctuated through very narrow limits, at least for such bodies as native 

 and vulcanized india-rubber, and therefore give for very extensive ranges of speed of 

 impact a thorough verification of Newton's experimental law ; viz. the constancy of the 

 coefficient of restitution for any given impinging bodies. This had, however, been long 

 ago carefully tested by the elaborate experiments of Hodgkinson.* There was, it is 

 true, a slight falling off for the very high speeds, and likewise for the very low : as 

 will be seen from the table (p. 397) which follows the experimental results. The first 

 may be due in part to a defect in the apparatus, the second will be accounted for below. 



The approximate constancy of e, for all relative speeds, proves merely that the force 

 of restitution is, at every stage, proportional to that required for compression. We must 

 therefore look to the values of the total distortion, or to those of the duration of impact, 

 for information as to the relation between the distortion and the force producing it. The 

 equation of motion during the compression is, say, 



Mx = Mg-F-f\x) (1) 



* British Association Report, 1834. 



