438 DE W. PEDDIE ON 



ami the total stress due to such groups is 



r(8-vrd)-foO-2Trrdr=27rka*(~a8-~a 2 &) (2) 







The total force tending to diminish the torsion is therefore 



| dfcNa 2 (a0) - IwWf 2 - f __L_ l( a 0) 2 . 

 3 4 L ■j.m^m- 1)J 



The single force which, acting at the distance a from the axis, would equilibrate this is 



l7rkNa 2 (a6)-^kva 2 



2m' ! (?n-l)J 



= ±7rWa 2 (a6) -lirJcva^ -\.( a ^) 2 ( 3 ) 



Hence f/*e deviation from Hooke's Law is represented by a negative term, involving the 

 square of the distortion, provided that the quantity v is constant. 



But v is the rate at which groups break down per unit change of distortion. Thus 

 (3) gives the theoretical deviation from Hooke's Law ivhen the range of distortion at 

 which a group breaks down is, on the average for all groups, uniformly distributed 

 over all possible ranges. 



If v were zero there would be no internal loss of energy in the wire ; and, if the wire 

 were once set in oscillation, the oscillations would, so far as this cause is concerned, con- 

 tinue for ever without any loss of amplitude. If v is very small, the difference between 

 the quantities of energy stored up in the wire in two successive maximum twists is 

 practically proportional to ydy/dx, where y is the scale-reading and x represents number 

 of oscillations, since Hooke's Law is nearly obeyed ; and we can easily prove (see below) 

 that the loss of energy in an outward oscillation is proportional to the cube of the 

 distortion. Also, since, by our fundamental assumptions, every group which broke down 

 at a certain stage in the outward motion breaks down again at the same point in the 

 inward motion, the total loss of energy, in the form of heat, in the inward motion to the 

 zero is equal to that in the outward motion from zero. Hence we get — bdy = fdx, 

 which gives 



y(x + a) = b . 



This is, as we have seen, precisely the equation which was found experimentally to con- 

 nect range of oscillation with number of oscillations when the wire is greatly fatigued. 

 Tf, therefore, our theoretical assumptions correctly represent the physical conditions, the 

 effect of great fatigue is to produce averagely uniform distribution of breaking range 

 over all p>ossible values. 



