Davis: Present Status of Integral Equations 27 
approximation, between the couple of torsion, P. and the 'angle of 
torsion, W, is given by the linear equation, 
W = k P , (12) 
where k is a physical constant. 
But it is reasonable to suppose that W does not depend merely 
upon the present moment of torsion, but upon all preceding ones as 
well. The elastic body has experienced fatigue from previous 
distortions and, in this way, has inherited characteristics from the 
past. We may express this analytically by saying that W(t) is a 
function of a line. 
When W(t) is developed according to the calculus of such 
functions, equation (12) is replaced by the integral equation, 
W(t) k P(t) + P K(t,s) P(s) ds , 
J to 
where K(t,s) is the coefficient of heredity. 
Assuming that W(t) and P(t) are both periodic functions of the 
time with the same period, Volterra shows that the coefficient of 
heredity is then of the form, 
K(t,s) = K(t-s). 
Experimental evidence for the existence of a true hereditary 
coefficient has been sought for, but results so far obtained are not 
conclusive. 
Here again as in the work of Hilbert we see integral equations 
in their application superseding differential equations as a tool of 
greater power and generality. How far these new equations are 
destined to replace the methods which have been standard in 
mathematical physics and astronomy for two centuries is one of the 
interesting questions to be answered by the future of mathematical 
research. 
