RECENT ADVANCES IN SCIENCE 353 



Sir T. Muir, in a note on compound determinants expressible 

 as simple determinants {Quart. Journ., 48, 1920) works on the fact 

 recently published in the Proc. Math. Soc. Edin., 36, by Professor 

 Whittaker that certain compound determinants are expressible 

 as simple determinants, and obtain some more general results. 



Sir T. Muir contributes a seventh list of writings on deter- 

 minants to the Quart. Journ., 49, 1920. This list covers the 

 writings on the subject up to the end of 191 9. 



Frank Morley {Ann. Math., 22, 1920) discusses the relations 

 between the centroids, symmedian points and other points 

 attached to a triangle, in the case of four triangles whose vertices 

 are formed by the omission one at a time of four given points 

 which lie on a circle. 



An addition has been made to the series of Cambridge Tracts 

 in Mathematics and Mathematical Physics. R. H. Fowler, in 

 " The Elementary Differential Geometry of Plane Curves " 

 (Tract number 20), presents a precise account of the elementary 

 differential properties of plane curves. No suitable connected 

 treatment of the subject has hitherto been available in English. 



Feuerbach's Theorem states that the existence of a circle 

 (the nine point circle) which touches the inscribed circle of a 

 triangle internally and each of the three described circles exter- 

 nally. This theorem is extended to apply to triangles formed 

 by arcs of circles. 



T. C. Lewis {Amer. Journ. Math., 42, 1920), in a paper entitled 

 " Is there an Analogue in Solid Geometry to Feuerbach's 

 Theorem?" considers this question put by L. Coolidge in his 

 Treatise on the Circle and the Sphere, and shows that the analogue 

 for three dimensions does not in general hold. 



L. E. Wear, in a paper on " Self-dual Plane Curves of the 

 Fourth Order " in Amer. Journ. Math., 42, 1920, discusses, in the 

 case of quartic curves, those curves called by Appel auto- 

 polaire, which are identical with their reciprocals. They turn 

 out to be the limagon and the degenerate case of two conies. 



R. W. Winger, in a paper " On the Satellite Line of the 

 Cubic " in the Amer. Journ. Math., 42, 1920, gives the explicit 

 equation of the satellite, for the general cubic in canonical 

 form. 



E. Cartan {Bull. Math. Soc. France, 47, 1919) in a paper 

 entitled " Sur les vari6t6s de courbure constante d'un espace 

 euclidien ou non-euclidien," discusses the general case of p 

 dimensions. His results with respect to 3-dimensional 

 Euclidean space were given in a series of communications to 

 the Comptes Rendus (167, 191 8). 



G. H. Hallett, Jr. {Ann. Math., 21, 1920) deals with Euclidean 



