150 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 11, NO. 7 



in which the hysteresis loop has an invariable form; or the damping 

 of free vibrations. For example, the displacement-load diagram for 

 a given method of loading, % = funct (X), and the hysteresis loop can 

 at once be found by eliminating the time between the load-time curve 

 and the displacement-time curve. 



This formulation need not be restricted to elastic lag phenomena. 

 It applies to all irreversible effects measured by some quantity x 

 whose value is fixed by the history of some other quantity A'. Thus 

 X might be termed the generalized displacement and X the generalized 

 load. The latter is a more flexible concept than generalized force, 

 because the product Xdx need not represent work. Table 1 shows 

 some of the possible applications of this method. 



TABLE 1. — Irreversible Time Effects 



Phenomenon 

 Calibration of aneroid barom- 

 eter 

 Elastic after-effect with torsion 

 Magnetic hysteresis 

 Residual charge of condenser 

 Action of selenium cell 

 Hysteresis of thermometer glass Change of temperature 



Generalized load, X 

 Change of air pressure 



Torque 



Magnetizing force 



E. M. F. 



Intensity of incident light 



Generalized displacement, x 

 Deflection of instrument 



pointer 

 Angle of twist 

 Induction 

 Quantity of charge 

 Change of resistance 

 Change of volume 



In what follows the terms load and displacement will be used for 

 short to designate X and x respectively, but it is to be understood 

 that all quantities retain their most general significance. The analysis 

 will be confined to the determination of y as a function of /, because 

 X can easily be obtained from y by equation (1). 



Dimensional theory. Let the physical constants needed for specify- 

 ing the irreversible properties of a body be represented by Ci, C^, . .C^ 

 and let X be the load at time r. Suppose while t varies from to t 

 the load passes through a maximum range R. Then the load history 

 can be specified by R, t, and the geometrical shape of a diagram having 

 X/R and r/t for coordinates. Therefore 



y - funct {R, t, Cr,C2,..C:) (2) 



in which the form of the function is unknown, but the same for all 

 processes with geometrically similar load diagrams. As (2) is a 

 qualitatively complete physical equation, it is subject to the usual 

 methods of dimensional reasoning. 



This leads to several interesting possibilities, notably the reduction 

 of the number of independent variables confronting the experimenter, 

 and the prospect of predicting effects outside the limit of direct mea- 

 surement by observations on physically similar models. 



