PROFESSOR TAIT ON IMPACT. 237 



We now write the equation of this part of the branch in the form 



r = A + B(0 + xf ; 



the numerical values of A, B, x being inserted, after x has been reduced to radians, and 

 B modified accordingly. The equations, in this final form, are printed below — each with 

 the data from which it was obtained. (A fine protractor, by Cary, London, reading to 

 one minute over an entire circumference, belongs to the Natural Philosophy Class 

 collection of Apparatus ; so that it was found convenient to deal with degrees in all 

 measurements of angle, and in the bulk of the subsequent calculations : — the results being 

 finally reduced to circular measure.) 



In the Tables below, after the data (enumerated above) from each experiment, come 

 the equations of the successive parts of each trace in order. In these, /3 V f3 2 refer 

 respectively to the rise and fall due to the first rebound ; 71,72 to the second rebound, 

 &c. 



To test the formulae thus obtained, other radii were measured, as far as possible from 

 those already employed, say for instance [20], [60], [ — 20], [ — 60], &c. These measured 

 values, and the corresponding values calculated from the equation (before reducing to 

 circular measure), are also given below. The agreement is, in most cases, surprisingly 

 close ; and shows that the assumption of nearly constant friction cannot be far from 

 correct. 



The whole of the above statement presupposes that the adjustments have been so exact 

 that the line of fall of the needle-point passes accurately through the centre of the disc. 

 On a few occasions, only, it was not so : — but the necessary correction was easily calculated 

 and applied, by means of the trace preceding the first impact ; even if the trace of the 

 first rebound did not reach to the level of the centre of the disc. In fact, if we wish 

 to find the curve which would have been traced on the disc had the adjustment been 

 perfect, it is easy to see that we must draw from each point of the trace a tangent to 

 the circle described about the centre of the disc so as to touch the true line of fall. The 

 position of the centre of the disc, relatively to the point of contact of this tangent, is 

 the same as that of the true point, relatively to the actual point, of the trace. This 

 applies, of course, to all parts of the trace, including the datum circle. 



In the special trace which has been selected for photolithography as an illustration 

 (see Plate) this adjustment is markedly imperfect ; much more so than in the worst of 

 the others. The path of the tracing-point passed, in fact, about 3 mm. from the centre 

 of the disc ; while, in the worst of the other cases, the distance was not more than half 

 as great. But this very imperfection serves to enable the reader to follow without 

 any difficulty the various convolutions of the trace. The measurements and reductions, 

 obtained from this specially imperfect figure, agree wonderfully with those obtained 

 from the best traces. It would only have confused the reader had we selected one of 

 the latter for reproduction, since each of them contains the record of four experiments — 

 i.e., it contains four times as much detail as does the trace reproduced. 



