236 PROFESSOR TAIT ON IMPACT. 



and the values tabulated satisfy these conditions. Thus the somewhat precarious assump- 

 tion as to the circumstances of restitution is, so far, justified. 



6. The tangents of the inclination of the trace to the radius of the datum circle 

 drawn to the intersection of these curves before and after impact . . . Aj, A 2 . 



These values were determined directly by drawing tangents to the trace ; and 

 indirectly by calculation from the equation of each part of the trace. The agreement of 

 the observed (o) and calculated (c) values is satisfactory. 



Attempts to form the equation of the part of the trace made before the first impact 

 were not very successful, as the available range of polar angle was small, and the radius 

 vector increases rapidly for small changes of that angle. Hence the calculated value of 

 Ax was obtained simply as the ratio of the tangential and radial speeds of the tracing 

 point at the moment of its first crossing the datum circle. This was taken as 



R&) R , 



— r- = tttt-^tt nearly. 



s/tyK 36-5N • J 



In this numerical reduction H is taken as 4 feet, i.e. 1219 mm.; and the full value 

 of g is employed, as we do not know the amount by which friction diminishes it, the 

 contact of the tracing-point with the disc coming about only during an uncertain portion 

 of the lower range of the fall ; while it is not possible to estimate with any accuracy the 

 effect of the impact on the trigger. The calculated value of the tangent will therefore 

 always be too small, but (since the square-root of the acceleration is involved) rarely by 

 more than 1 per cent. On the other hand, the graphic method employed for the direct 

 measurement of this tangent usually exaggerates its value. 



7. The ratios of these pairs of tangents — i.e., the values of the coefficient of 

 restitution .............. e. 



The equation of each distinct part of the trace (alluded to in 6. above) was found 

 thus : — The minimum (or maximum) radius-vector was drawn approximately for each 

 separate free path, and other radii were drawn, two on either side of it, making with it 

 convenient angles : — usually 40°, 80°, —40°, —80°, or such like. The notation employed 

 below for the measured lengths of these radii- vectores is simply square brackets enclosing 

 the value of the angle-vector, thus : — 



[80], [40], [0], [-40], [-80]. 



If x be the angular error introduced in the estimated position of the minimum radius, 

 we determine it, as well as the A and B of the equation of the corresponding half of the 

 branch of the curve in question, from three equations of the very simple form 



[0] = A + Ba 2 , 

 [40] = A + B(40 + xf, 

 [80] = A + B(80 + xf, 



(which may be made even more simple for calculation by putting y for 40 + x). The 

 assumed initial radius was in most cases so near to the minimum that very little difference 

 was found between [0] and A ; x being usually very small. 



