APRIL 4, 1921 hersey: irreversible time effects 153 



This is not an integral equation, for there is nothing unknown under 

 the integral sign. It has sometimes been thought that integral equa- 

 tions would be indispensable in solving "heredity" problems, but such 

 does not appear to be the case except when treating inertia. 



The fact that the constants in the drift function may depend on the 

 load could be shown explicitly by writing F as a function of two ar- 

 guments, t — T and A'. Since the drift constants are sensitive to 

 temperature change, this consideration has special importance for 

 problems of thermal hysteresis where the load is itself a temperature 

 change. It may often be sufficient to express F as the product of two 

 factors, one of which, Fo , is independent of A', so that 



Fit - r) =f{X).Fo(t - r) 



and 



t 



y = r^'/(A')/o(^ - r)dr (12) 



< dr 



For example, if p' denotes the fractional change in B with respect to load, 



B^BoiX + ^X) (13) 



so 



/(A) = 1 + ^A 



For vanishingly small loads /(A') approaches unity and (12) reduces 



to von Schweidler's formula for residual effects in dielectrics. 



When f{X) is constant, the integration of (12) by parts gives 



y = j X\f^{t — r)(ir in which \p{t — r) denotes the first derivative of 



o 



the drift function Fit — r), and in which it is understood that A' = 

 when r ^ 0. Writing co in place of / — t this reduces to 



00 



y = j AXco)J« (15) 



o 



which is Boltzmann's equation for the elastic after-effect in torsion 

 wires. Boltzmann's formula is therefore a special case of (12) above, 

 and a physical interpretation has been found for his arbitrary function 

 ^(w) by identifying it as the slope of the drift curve. 



Definition of ideal irreversibility. In all cases where some drift is 

 generated whenever the load on a body changes, this fact alone neces- 

 sitates qualitatively the existence of all the remaining irreversible 

 effects such as the familiar hysteresis loop. Hence, it is of interest 

 to compute the amount of each effect which would accrue from the 

 simple addition of elements of drift even where it is not expected that 

 the whole effect can be attributed to drift. In order to establish 



