548 SCIENCE PROGRESS 



M. J. M. Hill (Proc. Lond. Math. Soc. 191 7, 16, 219-72) 

 gives a classification of the integrals of linear partial differential 

 equations of the first order according to the values of the 

 Jacobian. 



R. G. D. Richardson (Trans. Amer. Math. Soc. 191 7, 18, 

 489-518) develops a new method by which the well-known 

 results about boundary problems for the ordinary and elliptic 

 partial differential equations can be deduced, and then attempts 

 an application of this method to an investigation of the facts 

 concerning the new boundary problem for the hyperbolic 

 equation. 



G. A. Pfeiffer (ibid. 1 85-98) shows the existence of divergent 

 solutions of the functional equation which is fundamental in 

 the problem of the conformal mapping of a curvilinear angle, 

 and shows that these solutions have a significance inherent to 

 the mapping problem referred to. 



F. Riesz (Acta Math. 1916, 41, 71-98) considers the inverse 

 problem for a certain class of linear transformations of con- 

 tinuous functions, and also an application to Fredholm's 

 integral equation. The object of the paper is not so much to 

 obtain new results about linear functional equations as to test 

 an exceedingly elementary method. By far the most important 

 conception used is one introduced by Frechet, and the limita- 

 tion to continuous functions is not essential in the paper. 



Geometry. — J. R. Kline (Trans. Amer. Math. Soc. 191 7, 18, 

 177-84) shows that the converse of the theorem of R. L.. 

 Moore (ibid. 191 6, 17, 132-64) concerning the division of a plane 

 by an open curve holds in spaces satisfying a certain set of 

 axioms formulated by Moore, and this in certain spaces which 

 are neither metrical, descriptive, nor separable. The converse 

 theorem of the analogous theorem of Jordan (1893) f° r simple 

 closed curves was first formulated by Schoenflies in 1902, who 

 made use of metrical properties in his proof. A different proof 

 was given by Lennes in 191 1, and this proof was commented 

 upon by Moore. 



A. J. Kempner (Amer. Math. Monthly, 191 7, 24, 317-21) 

 gives, after A. Reymond (191 6), a simple graphical method of 

 determining whether a given integer is a prime number or 

 not. This simple relation between projective geometry and 

 the theory of numbers is, says Kempner, really only an analogy 

 without deeper import, 



