270 SCIENCE PROGRESS 



theory could be transformed into the Laplacean form of equation 

 in three variables. 



In continuation of the well-known work of Schwarz on the 

 fifteen cases of the hypergeometric equation which admit of 

 algebraic integration, in which the conformal representation of a 

 curvilinear triangle is discussed, J. Hodgkinson (Proc. Lond. 

 Math. Soc. 1 91 6, 15, 166) discusses the conformal representation 

 of the various triangles bounded by the arcs of three intersecting 

 circles. 



G. A. Bliss (Trans. Amer. Math. Soc. 1916, 17, 195) in- 

 vestigates Jacobi's condition for problems of the calculus of 

 variations in parametric form. 



Geometry. — In April 191 5 Dr. R. L. Moore presented to the 

 American Mathematical Society a paper on the foundations of 

 plane analysis situs, and these researches, when expanded 

 (Trans. Amer. Math. Soc. 1916, 17, 131), give three systems of 

 axioms each of which systems is sufficient basis for a con- 

 siderable body of theorems in the domain of plane analysis 

 situs on what may be roughly termed the non-metrical part of 

 plane point-set theory (cf. Science Progress, 10, 433 and 616 ; 

 11, 92). The axioms of each system are stated in terms of a 

 class of elements called points and a class of point-sets called 

 regions. It may be mentioned that, on p. 136, the author cites 

 Veblen's proof of 1904 of the equivalence of the Heine-Borel 

 process and the Bolzano-Weierstrass process, but no mention is 

 made of the proof which the present writer gave soon afterwards 

 of the non-equivalence of these processes. In May 1916, Moore 

 (Proc. Nat. Acad. Sci., Washington, D.C., 1916, 2, No. 5 ; Nature, 

 191 6, 97, 395) set up two systems of postulates for plane analysis 

 situs based on the fundamental notions of point, limit-point, 

 and regions of certain types. Each set is sufficient for a large 

 number of theorems. 



G. N. Watson (Proc. Lond. Math. Soc. 1916, 15, 227) in- 

 vestigates a theorem of analysis situs somewhat similar to 

 Jordan's well-known theorem that a regular closed curve 

 possesses an interior and an exterior. Watson's theorem is 

 that when a simple closed curve with a continuously turning 

 tangent is given, if i/r be the angle between the tangent at a 

 variable point P of the curve and the it-axis, then yjr changes 

 by 27r as P describes the curve. 



In a paper read to the Royal Society on May 18, 1916, Col. 



