RECENT ADVANCES IN SCIENCE 195 



41, 123-32) has a note on seminvariants of systems of partial 

 differential equations. 



On ordinary and partial differential equations, cf. F. A. 

 Willers (13), J. B. Pomey (36), G. Hamel (27), O. Perron (24), 

 E. Hilb (19), R. Gamier (31, 33), T. Hayashi (49), L. Fejer (25), 

 G. Giraud (35), and H. Liebmann (30). On the calculus of 

 variations, cf. W. Blaschke (25), and K. Boehm (16) ; on 

 difference equations, cf. J. Horn (25) ; and on integral equa- 

 tions, cf. L. Lichtenstein (27), J. Horn (13), T. Lalesco (32, 34), 

 and E. Picard (32). 



Geometry. — A. Emch (Amer. Math. Monthly, 191 9, 26, 194- 

 201) gives a long and detailed review of Veblen and Young's 

 Projective Geometry (cf. Science Progress, 191 9, 13, 669). 



S. M. Ganguli {Bull. Calcutta Math. Soc, 191 8, 9, 1 1-8) 

 introduces a new method of studying inclinations of spaces of 

 any given number of dimensions less than n which are con- 

 tained in a space of n dimensions. A systematic algebraic 

 treatment — as distinguished from geometrical points of view 

 — of the subject is attempted on simple principles which are 

 applicable only to Euclidean spaces. 



R. Vythynathaswamy (Journ. Indian Math. Soc, 191 9, 11, 

 46-9) gives some theorems on conies with vanishing 0, ©' '. 

 P. Franklin (A mer. Math. Monthly, 191 9, 26, 146-51) discusses 

 the problem in solid geometry analogous to that of the con- 

 struction of a circle in a plane from three elements (points, 

 tangent lines, or tangent circles), which is the determination 

 of a sphere from four elements (points, tangent planes, or 

 tangent spheres), and is greatly simplified by the use of 

 certain principles given. W. Sensenig (Amer. Journ. Math., 

 1919, 41, 1 1 1-22) discusses the invariant theory of involutions 

 of conies. 



C. E. Cullis (Bull. Calcutta Math. Soc, 191 8, 9, 23-42) shows 

 (1) that rotations of a rigid body about concurrent axes fixed 

 in space can be replaced by the same rotations about the same 

 axes moving with the body, provided that the order in which 

 the rotations are applied is reversed ; (2) that the polar of a 

 spherical polygon admits of a unique definition and has the 

 same uses as the polar of an ordinary spherical triangle ; 

 (3) that these results can be used to obtain complete general- 

 isations of Rodrigues' and Sylvester's theorems regarding 

 rotations. 



