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Indiana University Studies 
trade, the applications of E. Baticle to problems in the stability of 
structures, C. Charlier’s investigation of terrestrial refraction and 
the composition of the atmosphere, and the discussion of T. Hayashi 
on the theory of diffusion of mixed gases indicate the broad class of 
problems to which the new theory can be applied. 
The latest application to be made of integral equations is found 
in the mathematical theory of economics. This study has been 
initiated by G. C. Evans , 478 C. F. Roos , 491 and H. Hotelling , 492 and 
gives promise of laying the foundations of a new and more exact 
science of economics. To quote from Professor Evans: “There 
is not only an opportunity for mathematicians in economics, but 
even a duty; and on the mathematicians in an unusual degree lies 
the responsibility for the economic welfare of the world.” Hotelling 
has founded a theory of depreciation which leads to a Yolterra 
equation of second kind with a kernel of the form K ( x , t ) = 
A promising application of integral and integro-differential 
equations has been made by Yolterra in the study of problems in 
so-called hereditary mechanics, which includes the phenomena of 
elastic and magnetic hysteresis. The name and definition are due to 
E. Picard, whom we quote as follows:* 
In all this study [of classical mechanics] the laws which express our ideas 
on motion have been condensed into differential equations, that is to say, 
relations between variables and their derivatives. We must not forget that 
we have, in fact, formulated a principle of non-heredity , when we suppose that 
the future of a system depends at a given moment only on its actual state, or 
in a more general manner, if we regard the forces as depending also on velocities, 
that the future depends on the actual state and the infinitely neighboring- 
state which precedes. This is a restrictive hypothesis and one which, in 
appearance at least, is contradicted by facts. Examples are numerous where 
the future of a system seems to depend on former states: here we have heredity. 
In some complex cases one sees that it is necessary, perhaps, to abandon 
differential equations and consider functional equations in which there appear 
integrals taken from a distant time to the present, integrals which will be, in 
fact, this hereditary part. The proponents of classical mechanics, however, 
are able to pretend that heredity is only apparent and that it amounts merely 
to this, that we have fixed our attention upon too small a number of variables. 
But the situation in this case is just as it was in the simpler one, only under 
conditions that are more complex. 
The following example from Yolterraf will help to clarify this 
idea. We know from elementary physics that the relation, to a first 
*La Mecanique classique et ses approximations successives. Revista di Scienza, vol. 1 (1907), 
pp. 4-15. In particular, p. 15. 
t'Ref. 19, pp. 138-139, and 150. 
