io SCIENCE PROGRESS 



M. Frechet and J. Pierpont (Bull. Amer. Math. Soc. 1917* 

 23, 172-5 ; cf. Science Progress, 1916, 11, 94) continue their 

 discussion on Pierpont 's definition of integrals. 



S. Pollard (Proc. Lond. Math. Soc. 1916, 15, 336-9) inquires 

 how far the results of the older theory of Fourier's series, which 

 is concerned with direct investigations into convergence, based 

 mainly on Dirichlet's integral, can be deduced from the more 

 modern results as to the summability of the series, based mainly 

 on the properties of Fejer's and Poisson's integrals. Jordan 

 has obtained a result in this order of inquiries, but his criterion 

 deals with convergence throughout an interval ; Pollard deduces 

 Dini's more delicate criterion, dealing with convergence at a 

 point, from Fejer's theorem. 



G. H. Hardy and J. E. Littlewood (Proc. Nat. Acad. Sci., 

 Washington, D.C., 191 6, 2, No. 10) give a remarkable trigono- 

 metric series which is never convergent or summable for any 

 value of its argument, and is thus not a Fourier's series ; and 

 a function which has no finite differential quotient for any value 

 of its argument. 



W. B. Ford (Amer. Journ. Math. 19 16, 38, 397-406) supple- 

 ments some researches of Paul du Bois-Reymond, Dini, Hobson, 

 and Lebesgue by pointing out a noteworthy class of functions 

 <f> such that the limit when n is infinite of the integral 



I 



b 

 f(x)4>(n, x — a)dx 



should take the well-known Fourier mean value of the arbitrary 

 f(x) at x = a, it being assumed that f(x) satisfies suitable condi- 

 tions in the neighbourhood of x = a. 



Sir Ronald Ross and Miss H. P. Hudson (Proc. Roy. Soc. 

 February 1, 191 7) give the second part (cf. Science Progress, 

 1 91 6, 11, 93-4) of Ross's application of the theory of proba- 

 bilities to the study of a priori pathometry. This part is 

 occupied with the construction and examination of a number of 

 hypothetical epidemics on the basis of the equations of the first 

 part. The conclusions suggest that the rise and fall of epi- 

 demics may be explained by the general laws of happenings as 

 studied. At the same meeting J. Brownlee investigated the 

 periodicity of measles epidemics in London from 1703 to the 

 present day by the method of the periodogram. 



The problem of Lagrange in the calculus of variations is 



