94 SCIENCE PROGRESS 



method seems to have been first used by Sir Ronald Ross in 

 previous papers, and a more advanced and general develop- 

 ment of this previous work is now given by him (Proc. Roy. 

 Soc. 1 91 6, A, 92, 204), in which the curves are theoretically 

 obtained when we suppose that the infectivity-ratio remains 

 constant or proportional to the number of individuals already 

 affected, while simultaneously some of these are constantly 

 losing immunity, and also both the affected and the unaffected 

 groups are subject to special rates of birth, death, immigration, 

 and emigration. There are no great mathematical difficulties, 

 and the differential equations are all solved by known methods, 

 but the application is an exceedingly interesting and important 

 one, both theoretically and practically, of the theory of proba- 

 bility. Cf. also an article by M. Greenwood in Nature (1916, 

 87, 243). 



Analysis. — Prof. W. H. Young (Proc. Lond. Math. Soc. 

 1 91 6, 15, 35) continues his investigations of 19 10, 191 2, and 

 191 3 on derivates of the integral of a summable function. 



A controversy between Profs. M. Frechet and J. Pierpont 

 (Bull. Amer. Math. Soc. 1916, 22, 295 and 298) on the new 

 definition of Lebesgue integrals given by Pierpont in the second 

 volume of his Theory of Functions of Real Variables is of some 

 interest. 



G. H. Hardy (Proc. Lond. Math. Soc. 191 6, 15, 72) proves 

 at length a theorem of consistency for summable series enun- 

 ciated without proof in The General Theory of Dirichlet's Series, 

 by Hardy and Marcel Riesz, reviewed in this quarterly for 

 January 1916, p. 500. 



Motoji Kuniyeda (Proc. Lond. Math. Soc. 19 16, 15, 128) 

 proves a theorem for a general series of orthogonal functions 

 which is analogous to a theorem proved by Hardy and Little- 

 wood in 1 91 3, and is connected with Lebesgue 's extension (1905) 

 of Fejer's theorem of 1904 about the summability of Fourier's 

 series. Kuniyeda used the method which Hobson used in a 

 paper referred to in this quarterly (191 6, 10, 619) for proving 

 a theorem of Weyl ; but the theorem of Weyl and Hobson is 

 not used, so that it can be deduced as a corollary of Kuniyeda 's 

 theorem. 



O. D. Kellogg (Amer. Journ. Math. 1916, 38, 1), in a paper 

 on the oscillation of functions of an orthogonal set, finds the 

 condition that a linear function of n orthogonal functions can 



