Pebruaky 12, 1915] 



SCIENCE 



247 



is merely a sign that the instantaneous condi- 

 tions are insufficiently known. The other 

 point of view is that the history can not 

 he replaced hy the consideration of contem- 

 poraneous elements; in other words, that 

 a finite number of present elements can 

 not replace the infinity past instants in the 

 determination of the state of the system. The 

 question as to the presence of heredity effects 

 in physical phenomena is, as Volterra points 

 out, of the same character as the old New- 

 tonian question of action at a distance. In 

 fact, if we take account of the theory of rela- 

 tivity and the four-dimensional space-time 

 space, the two questions meet. 



Such questions are important if we try to 

 reduce our system of the world to one that is 

 entirely kinetic. In that case we must get rid 

 of the " coefficients of heredity " by explaining 

 them in terms of concealed motions. It may, 

 however, be impossible completely to reduce 

 physical phenomena to a finite number of ele- 

 ments, no matter how described in terms of 

 functions and variables, without exceeding the 

 time limit for speech; and one method may 

 not be more " fundamental " than another. 

 But regardless of our attitude towards the two 

 aspects of the question, or our opinion of the 

 practical value of making such distinctions 

 on the ground of " reality " or " truth," we 

 can not in any case deny the value of the 

 analysis that enables us to take account of 

 such a thing as the history of the system. 



Let us turn now to two subjects, elasticity 

 and electricity, where this analysis seems to 

 be usefully introduced. In the usual treat- 

 ment of elasticity, we have Hooke's law, con- 

 necting the deformations and tensions of the 

 system; in electricity the induction and dis- 

 placement are also connected by linear rela- 

 tions. If now we assume that the tension at 

 any time depends linearly not only on the de- 

 formation at that time, but also on the de- 

 formation at all previous times, we can intro- 

 duce this fact into our equations by adding, 

 in our expression of Hooke's law, a term in 

 the form of an integral, whose integrand rep- 

 resents the contribution to the tension at a 

 time t, due to a deformation acting at a time 



T through an interval of time dr. In this 

 way, Hooke's law becomes an integral equa- 

 tion, or a system of integral equations, and 

 the differential equations that determine the 

 deformations or the tensions become integro- 

 differential equations. In a similar way, in- 

 tegro-differential equations are introduced 

 into the subject of electricity. 



The study of the methods of integro-differ- 

 ential equations, their solutions, and their 

 applications to the subject of hysteresis, or 

 heredity, form the subject matter of the book 

 from Chapter V. on. In connection with the 

 relative importance which the theory of this 

 subject has assumed in the presentation of 

 Professor Volterra, we may remember that in 

 the ease of elasticity it seems to have received 

 important experimental verification in the 

 work of our American physicists. Professor 

 Webster and Dr. Porter. 



A detailed analysis of the contents of the 

 book is unnecessary. Some points, however, 

 should be given special mention, because of 

 their universal interest. Chapter TV. is de- 

 voted to functional equations in general; that 

 is, to implicit functions of curves. Theorems 

 are obtained which correspond, first to the in- 

 version of an analytic function, and second to 

 the more extensive theorem on the determina- 

 tion of implicit functions in general. In fact 

 it may be noticed that the theorem might be 

 given in such a form as to include the ordi- 

 nary theorem on implicit functions as a spe- 

 cial case, although with respect to the scope 

 of the book sudh a generalization would be 

 trivial. The condition for the " closed cycle " 

 (Chapter VII.) deserves special attention, be- 

 cause of its relation to the problem of hered- 

 ity, which, as we have seen, is 'a central one 

 for the book. In this chapter, section 10 is a 

 first essay at a possible treatment of magnetic 

 hysteresis. Another interesting subject is the 

 application of permutable functions to the so- 

 lution of integro-differential equations. It is 

 in connection with this subject that are in- 

 troduced various new sorts of transcendental 

 functions, similar in a way to the exponential 

 function, the sine, and so on. The quality 

 of periodicity, which appears to be lacking, 



