441' DR W. PEDDIE ON 



by (7); the next outward motion, the in motion being stopped just short of the zero, 

 again obeys the law of force given by (9) ; and so on. By taking 2).^ instead of 2. i 



in equation (8) we get an expression for the angle of set in the first part of the out- 

 ward motion under these circumstances. 



We can easily get a simple graphical construction for the two extreme positions 

 of set. Plot forces as abscissas and angles as ordinates. Draw the Hooke's Lav/ line 

 as indicated by the first term of (3). Draw r also the parabolic curve given by (3), and 

 the parabolic curve indicated by the first two terms of (9). Take three-fifths of the 

 difference of abscissas of the Hooke's Law line and the former parabola at the ordinate 

 corresponding to the maximum angle 0, and plot it along the line of abscissae. The 

 ordinate drawn through the point so found intersects the two parabolas at points whose 

 ordinates are the extreme angles of set. The method is shown in fig. 1 5. 



The dotted curve in fig. 15 is the second parabola above referred to, the full curve 

 being the first. The position of set being taken as origin, the dotted curve does not 

 greatly differ from a straight line, the deviations at the larger forces being in the 

 direction of too great distortion. This result explains Wiedemann's observation (Philo- 

 sophical Magazine, vol. ix., 1880) that, after a wire has been tivisted a feiv times in 

 opposite directions alternately by a given couple, and is then twisted by increasing 

 couples in the direction of the last tivist, Hooke's Law is nearly obeyed, provided the 

 original couple is not exceeded, the slight deviations being in the direction of too 

 great tivist. 



In order to deduce the expression 



y n (x + a) = b 



as the more general relation connecting range of oscillation with number of oscillations, 

 we have only to assume that the quantity v, employed in the preceding investigation, 

 varies as a power of the strain. Take £ = r0/ (m+p) where m is a whole number and p 

 is a proper fraction ; and, instead of v, let us write 



\m+pj 



where v and m are regarded as constants. Each group which breaks at £ has, when it 

 breaks, potential energy |-&£ 2 , which is transformed into heat. Also each such group, 

 p varying from to 1, breaks m times. Hence the heat developed in the range to 0, 

 is, in the volume 2irrdr, 



(rey-n* rdr 



I \m+p) \m+p/ \m+p) 



V 



3 + P j» 2+ ' 4 (m+l) 8+ '' 



