6 SCIENCE PROGRESS 



The plane contour used by the author means any closed 

 plane curve regular in Osgood's sense, or composed of a finite 

 number of analytic arcs or straight segments. The polygon is 

 a special case. The Green's function in polars is 



G{r, 6) = -logr + u{r, d) 



where u satisfies Laplace's equation and has no singularities in 

 the region concerned. The solution of the resulting equations 

 can be obtained in a formal manner by integral equations, but 

 is only formal, and it is necessary to use the author's method 

 of direct solution of the equations, which, however, does not 

 often give series with a general coefficient capable of recognition 

 — this is not a matter of concern in the applications. The 

 square contour is selected as a convenient illustration of the 

 analysis. 



W. G. Simon, in the same number, discusses the solution 

 of some types of linear diff'erential equations in an infinite 

 number of variables, mainly with a view to proving the exist- 

 ence of certain forms of solution, chiefly exponential. His 

 starting-point is an existence theorem of von Koch, and a 

 generalisation of a theorem of Poincare regarding the develop- 

 ment of solutions of differential equations in power series of a 

 parameter yu,, when the functions appearing in the differential 

 equations are themselves power series in /i. Considerable dis- 

 cussion of some forms of infinite determinants is included. 



D. Buchanan, in the same Journal, discusses some inter- 

 esting cases of Periodic Orbits on surfaces of revolution, the 

 orbits being those of a particle under gravity, with the axis of 

 the surface vertical. 



F. Riesz {Acta Math., xlii, 3, p. 192) discusses Lebesgue's 

 integral. The memoir is a continuation of a note in the Comptes 

 Rendus of 191 2, in which the idea of the integral was introduced 

 independently of the theory of measure, following a suggestion 

 of Borel. The author's point of view is, however, quite different 

 from that of Borel. 



P. Levy [Acta Math., xlii, 3, p. 207) deals with Green's and 

 Neumann's functions. The main initial problems proposed are : 



(i) If ^3 is one of these functions, to determine 1/^3 in such 



a way that ^3 — i/^^ is a holomorphic function of the points 

 A and B. 



(2) To form a function -^^^ such that the difference ^J — ■f'^ 

 may be finite, with all its derivatives, to a given order. 



These functions -^ are treated like the Green and Neumann 

 functions in the sense that, once obtained, they are capable of 



