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preceding states, but also on the whole set of preceding states. 

 This has led to the investigation of what has been called 

 " hereditary mechanics," and the only mathematical instru- 

 ment for dealing with it is the modern functional calculus : 

 integral equations and integro-differential equations allow us to 

 solve the problems of hereditary mechanics and physics. The 

 best treatment of the subject is contained in Volterra's Fonctions 

 de lignes and Lemons sur les equations integrates et integro- 

 differentielles. M. Winter {Rev. de Metaphys. et de Morale, 

 1 91 6, 23, 268) gives an easily understood account of these 

 things, together with some philosophical considerations. 



In some researches on the kinetic theory of ions in gases, 

 F. B. Pidduck (Proc. Lond. Math. Soc. 191 6, 15, 89) started 

 from Hilbert's (1912) rigorous treatment of Boltzmann's equa- 

 tion, in which treatment of a certain integral was reduced to 

 a form resembling the left-hand side of an integral equation. 

 In one case Pidduck found an integral equation and solved it 

 numerically, apparently the first instance of numerical solution 

 of an integral equation as distinguished from a theoretical 

 solution. The chief use made of Hilbert's transformation is, 

 however, to simplify calculations which might otherwise have 

 been inextricable. 



Mandelstam showed in 191 2 that a certain problem in optics 

 leads to an integral equation, and pointed out, with Lord 

 Rayleigh, that the solution of the problem depends on the 

 evaluation of an integral whose integrand contains a Bessel's 

 function. G. Steic (Amer. Journ. Math. 191 6, 38, 97) evaluates 

 this integral. 



Geometry. — In an interesting address on the historical de- 

 velopment and the future prospects of the differential geometry 

 of plane curves, Prof. E. J. Wilczynski (Bull. Amer. Math. Soc. 

 1 91 6, 22, 317) concludes that the notions osculant and penoscu- 

 lant are the fundamental concepts of differential geometry. The 

 systematic investigation of the magnitudes, loci, and envelopes 

 determined by the various classes of osculants and penosculants, 

 and the relations which exist between them, makes up the 

 whole subject-matter of differential geometry. Differential 

 properties of a general curve are merely integral properties of 

 its osculants and penosculants. 



In a similar way to his generalisation of the notion of angle 

 (1906), G. A. Bliss (Amer. Journ. Math. 191 5, 37, 1) generalises 



