TORSIONAL OSCILLATIONS OF WIRES. 439 



The apparatus which was used in the experimental investigations was not suitable 

 for the purpose of testing the expression (3) directly in its application to the torsion of 

 wires. Table VII. has been drawn up for me by Mr P. S. Hardie, formerly Neil 

 Arnott scholar in the Physical Laboratory, to test the applicability to the bending of 



1 tars of the equation 



y = ax — bx 2 , 



where y represents distorting force and x represents distortion. The data used in the 

 calculation are some of those given by Hodgkinson and Fairbairn in the B. A. Reports, 

 1837. The columns headed x and y give observed values of these quantities ; the 

 columns headed y' give calculated values of y. The correspondence is extremely close, 

 in some cases remarkably so, when it is considered that any flaw in the homogeneity 

 of the material tends to introduce irregularities in the action under stress. Fig. 15 

 exhibits graphically the results in one case. The full curve represents a curve y — ax — bx 2 , 

 and the points on or near it are obtained from the experiments. The straight full 

 line in the diagram represents the Hooke's Law line y = ax. The coordinate, y = a 2 /4:b, 

 of the vertex of the parabola corresponds theoretically to the breaking stress. The 

 material always, as is to be expected, breaks at a smaller stress. 



We have now to investigate the inward motion. At any stage, all groups which 

 give rise to an inward force in the outward motion give rise to the same inward force 

 in the inward motion, provided that their last breaking-point has not been repassed. 

 On the other hand, those groups whose last breaking-point has been repassed do not exert 

 ;m inward force, but in general exert an outward force. Hence the inward force at any 

 stage on the inward motion to zero is less than the inward force at the same stage on 

 the outward motion. Thus we deduce at once from the theory the observed result that 

 the time of outward motion over a given range is less than the time of inward motion 

 over the same range. 



Let us suppose now that the angular distortion (f>, in the inward motion, has become 

 loss than half the maximum angular distortion 0. Every group which broke clown in 

 the outward motion is now exerting an outward force. In the volume 2-rrrdr, since we 

 are assuming that the breaking range of distortion for different groups is, on the 

 average, uniformly distributed over all possible values, all groups which broke first 

 between <p and 6 are now exerting on the average an outward force ^kr(6 — <f>). All 

 those which broke at a range less than <j> are now exerting an outward force which is 

 proportional to the distance between r(p and their last breaking-point on the inward 

 motion. To find the total value of this force, consider m£ = r<t> r (m — 1)£ 7 = rep. A group 

 which broke at 



m \ m-\ 



had its nearest breaking-point outside r(p at wi£". Its distortion is therefore m£'' - r<f> = 

 pr(j>j(m — l). Now, at the fixed point rcj), when %' ranges over £-£',£> takes all values 

 from to 1 uniformly, so that its average value is \. Hence we find that the outward 



