PURE MATHEMATICS 353 



spond to the tangent indicatrix, showing that all curves with 

 two given plane curves as indicatrices are geodesies on a surface 

 of which the geodesic geometry is Euclidean ; he examines in 

 elliptic space the analogues of Bertrand and Mannheim curves, 

 and he proves that there is a linear relation connecting the 

 curvature and torsion at a point of all curves of a complex 

 which touch one another at the point. 



In the first edition of the Theorie des Surfaces, Darboux 

 suggested the investigation of all algebraic curves which have 

 constant torsion ; various authors attempted the problem, and 

 a few years ago it was proposed as the subject of a prize essay 

 by the Paris Academic des Sciences. The prize was shared 

 by B. Gambler and G. Darmois, and the latter's essay is now 

 printed {Ann. de Toulouse, 11, 191 9, 67-189). 



W. Blaschke and K. Reidemeister (jahresber. d. Math. 

 Verein., 31, 1922, 63-82) give an account of the development 

 of affine geometry (see Science Progress, 17, 1922, 180) ; 

 the latest paper in Blaschke's series is by himself {Hamburg 

 Seminar, 1, 1922, 206-9) ; he improves Minkowski's inequality 

 for the " mixed " surface area of two surfaces. 



L. Berwald {Monatshefte Wien, 32, 1922, 89-106) extends 

 some of Blaschke's work to n dimensions. 



Other recent geometrical papers are : 



A. CoMESSATi, " Nuovo contributi geometrici alia teoria delle forme binarie," 



Rend. Lincei, 54, 192, 541-51. 

 C. Servais, " Surles courbes du quatri^me ordre et de la troisi^me classe," 



Mem. de Liege, 11, pt. 2, 1922, No. i. 

 L. J. Smid, junr., " Number of Circles touching Plane Curves defined by Repre- 

 sentation on Point Apace," Proc. Amst. Acad., 35, 1922, 221-2. 

 E. Veneroni, " Sulle congruenze [21] de coniche che appartengono ad 



inversioni spaziali," Rend. Lombardo, 55, 1922, 137-48. 

 G. ScHAAKE, " A New Method for the Solution of the Problem of the 



Characteristics in the Enumerative Geometry," Proc. Amst. Acad., 35, 



1922, 113-17. 

 C. A. Nelson, "Conjugate Systems with Conjugate Axis Curves," Tohoku 



Math. J., 20, 1922, 217-51. 

 H. LiEBMANN, " Eine charakteristische Eigenschaft der H-Netze (Ortho- 



gonalprojectionen von Haupttangentenkurven)," Math. Zs., 14, 1922, 



159-68. 



B. Gambier, " Representation conforme avec conservation des lignes de 



courbure et de la valeur absolue du rapport des rayons de courbure 

 principaux," Ann. ic. Norm. Sup., 39, 1922, 217-72. 



C. DE Jans, " Note sur les courbes trochoidales," Mhn. de Lidge, 11, pt. 2, 



1922, No. 2. 

 M. Weill, " Sur les courbes rectifiables," Bull. Soc. Math, de France, 50, 



1922, 42-62. 

 T. Takasu, " On Intrinsic Equations of Twisted Curves," Sci. Rep. Tohoku, 



11, 1922, 29-41. 

 B. VON KERiKjART6, " Kurvenscharen auf Flachen," Gott. Nach., 1922, 71-9. 

 R. FuRCH, " Orientierung von Hyperflachen im projektiven Raum," Hamburg 



Seminar, 1, 1922, 210-u, 



