352 G. F. Becker — Finite Flastie Stress-Strain Function. 



Turning now to the first member of (7), values of x 1 and y 1 

 appropriate to the case in hand must be substituted. Each 

 point of the unit cube during shear moves on an equilateral 

 hyperbola, so that if x , y Q are the original coordinates of a 

 point, x 1 y 1 = x y . For the corner of the cube, whose coordi- 

 nates are x and y, the path is xy = 1. !Now x 1 /x Q = x and 



2/i/2/o = V so tnat 



x 1 dy 1 — y 1 dx l = x y (xdy — ydx) 



If ip is the area which the radius vector of the point a?, y 

 describes during strain, it is well known that 2d<p = xdy — ydx 

 and, since in this case xy = 1, it is easy to see that 



2dip = 2c? Iny 



Since the quantities x and y refer to a single point, the sign 

 of summation does not affect them, and the first member of 

 (7) may be written 



d In 1/ ~ 

 dt 2 ° y ° 



Here one may write for m, pdx dy , where p is the constant 

 density of the body ; and since the substance is uniform, sum- 

 mation may be performed by double integration between the 

 limits unity and zero. This reduces the sum to p/2. 

 Value of a. Equation (7) thus becomes 

 dHny _ 2 MQ 

 dt 2 ~~ p ' Qn 



the second member being constant. Counting time from 

 the instant of release, or from the greatest strain, and inte- 

 grating y between the limits y = a and y = 1 gives 



MQ f 



Ina = — - . — 

 6w p 



It is now time to introduce the hypothesis that the vibra- 

 tions are isochronous. It is a well known result of theory 

 and experiment that a rod of unit length with one end fixed, 

 executing its gravest longitudinal vibrations, performs one 

 complete vibration of small amplitude in a time expressed 

 by &Vp/M. In the equation stated above t expresses the time 

 of one-quarter of a complete vibration or the interval between 

 the periods at which y — 1 and y — a. Hence for a small 

 vibration, t as here defined is *s/p/M. If the vibrations are to 

 be isochronous irrespective of amplitude, this must also be the 

 value of t in a finite vibration. Hence at once 



the same result reached in (5). 



