S4o SCIENCE PROGRESS 



not regard the position even then as satisfactory. The present 

 paper is based on a set of assumptions in terms of the notions 

 of point, region, and motion. The space which undergoes 

 the transformations (or motions) is not subjected in advance 

 to the conviction of being a number plane, and it is not supposed 

 in advance that the regions are in one-one correspondence 

 with portions of such a plane. Hilbert analysed the group of 

 transformations but not the space undergoing them. The 

 author's treatment subjects both to a simultaneous analysis. 

 M. E. Cartan, Bulletin de la Societe Mathematique de France, 

 xlvi. (1919), pp. 84-105, continues his researches on certain 

 hypersurfaces in spaces of five and more dimensions. 



M. D'Ocagne, Comptes Rendus, 170 (1920), pp. 1 6-1 71, 

 discusses the distribution of curvature round a point of a surface, 

 in relation to the Meusnier surface associated with the point. 

 F. Morley, Amer. Journ. of Math., xli. (191 9), pp. 279-283, 

 discusses the Luroth Quartic Curve, starting from Bateman's 

 theorem that the seven points which have the some polar line 

 as to a conic and a cubic give rise to a Luroth quartic. 



A. B. Coble, Amer. Journ. of Math., xli. (1919), pp. 243-266, 

 discusses the ten nodes of the Rational Sextic and of the Cayley 

 Symmetroid. It is proved that the theorem of Conner, in 

 the Journal for 191 5, regarding the intimate relation between 

 the nodes of the Sextic has its analogue with reference to the 

 Symmetroid under regular Cremona transformation in space. 

 W. P. Milne, Proc. Lond. Math. Soc. (2) xvii. (1919), pp. 274- 

 280, discusses the determinantal systems of Coapolar Triads 

 on a cubic curve. The paper is a sequel to one already noticed 

 a year ago. 



J. Hodgkinson, Proc. Lond. Math. Soc. (2) xviii. (1919)* 

 pp. 268-274, discusses a problem of conformal representation. 

 The curves dealt with form curvilinear triangles whose sides 

 have a real common orthogonal circle, and these are repre- 

 sented on the half plane of a new complex variable. The 

 present paper is designed to elucidate the conditions and 

 limitations of validity of the previous analysis. 



P. Boutroux, Comptes Rendus, 170 (1920), pp. 164-166, 

 considers further a class of multiform functions associated with 

 a differential equation of the first order. The equation is : 



zz\ — imz + 2x i -f- bx + c 



and the functions present an infinite number of branches and 

 singularities. The conditions of the formation of these func- 

 tions are reduced to a simple form. The functions are shown 

 to be automorphic in a new type. 



G. Valvion, Comptes Rendus, 170 (1920), pp. 167-169, 



