246 



SCIENCE 



IN. S. Vol. XLI. No. 1050 



SCIENTIFIC BOOKS 

 Logons sur les Fonctions de Lignes. Pro- 



fessees a la Sorbonne en 1912, par ViTO 



VoLTERRA, recueillies et redigees par Joseph 



Pekes. Paris, Gauthier Villars, 1913. 



The point of view of this book of Volterra's 

 is the systematic generalization of systems of 

 relations of simple type by means of a passage 

 from finite to infinite. We are already fa- 

 miliar with this procedure, in the subject of 

 integral equations, first in the work of Volterra 

 himself, suggested then in the work of Fred- 

 holm, and minutely worked out in the papers 

 of Hilbert, his associates and students. But 

 whereas perhaps Hilbert has limited himself 

 to a few aspects of the question and rigorously 

 justified the passage from finite to infinite, 

 considering the subject of forms in an infinite 

 number of variables as a subject for investiga- 

 tion in itself, Volterra has made wide applica- 

 tion of an heuristic device for the purpose of 

 obtaining results, which can then sometimes 

 more simply be justified by methods proper to 

 the new subjects themselves. This device is 

 as old as the idea of infinitesimals. 



After mentioning the familiar generaliza- 

 tions of this kind, of sum and product, Vol- 

 terra considers briefly the subject of the gen- 

 eralization of the multiplication of substitu- 

 tions, coordinate with the integration of linear 

 differential equations, and then devotes the 

 pages of the book proper to the generalization 

 of the idea of function of several variables. 

 This generalization involves the general prin- 

 ciples of the study of functional relations. 



We are concerned then, in the limit, with 

 the investigation of functions which depend 

 on an infinite number of variables — -in par- 

 ticular, on all the points of a curve, or on all 

 the values of another function throughout a 

 certain interval. The general method of mak- 

 ing such a study is by procedure from the 

 finite to the infinite. 



As an illustration of such a procedure, Vol- 

 terra cites, in the Introduction, a treatment of 

 the restricted problem of three bodies, by 

 the application of Cauchy's method. The mo- 

 tion of the small body, the only one not 

 known, can be determined by summing the 

 motions obtained by considering the larger 



bodies as temporarily fixed at various points 

 of their orbits, and proceeding to the limit as 

 these various points on each orbit are taken 

 closer and closer together. 



Another passage of the Introduction relates 

 to the definition of the derivative of a func- 

 tion of a curve, and is worth while quoting, 

 since in this ease the example is proper to our 

 subject itself. " If a quantity depends upon a 

 curve, we can study the effect produced on the 

 quantity by a variation of the curve. If this 

 variation is very small and limited to the 

 neighborhood of a point of the curve, we arrive 

 at the notion of derivative.^ For each point of 

 the curve we shall in this way have a derivative. 

 By superposing such variations of the curve, 

 made in all its points, we find the differential, 

 or variation, of the quantity, which will be ex- 

 pressed by means of an integral; in fact, since 

 a function of a curve is a function of an infi- 

 nite number of variables, the sum which ex- 

 presses the differential of a function of several 

 variables leads, by the passage to the infinite, 

 to an integral. 



"We can then take up the study of differ- 

 entials of higher order, and thus come to an 

 analytic development analogous to the Tay- 

 lor's series. The siiaple double and triple 

 sums, etc., which occur in the development of 

 a function of several variables, are replaced by 

 simple, double, triple, etc., integrals." 



The character of this analysis is thus shown. 

 Its purpose is to investigate the phenomena 

 of hysteresis and " evolution " or " heredity " 

 in physical systems — occurrences where the 

 state of the system is supposed to depend upon 

 the history of the system, i. e., to depend upon 

 the values of certain functions throughout all 

 previous instants of time. 



In regard to hysteresis and evolution, in 

 physics as in biology, we may adopt two dif- 

 ferent points of view. One possible stand- 

 point is that the future state of a system is 

 determined entirely by its state at a given 

 instant, and if the history of a system is used 

 in determining its subsequent behavior, that 



1 As the limit, under proper restrictions, of tlie 

 ratio of the variation of the function to the inte- 

 gral of the variation of the curve, in that neigh- 

 borhood. 



