354 SCIENCE PROGRESS 



and double elliptic geometries of three dimensions, the notions 

 of point and order being undefined. 



J. W. Alexander {Bull. Amer. Math. Soc, 26, 1920) proves 

 for the case w = 3, that every closed orientable n-dimensional 

 manifold can be represented on an w-dimensional hypersphere 

 as a Riemann space or generalised Riemann surface. The 

 extension to higher dimensions is perfectly automatic. 



The sixth of a set of memoirs by P. Koebe on conformal 

 representation appears in Math. Zeitschrift, 7, 1920. 



W. Burnside, in a memoir entitled " On Cyclical Octosection " 

 in the Camb. Phil. Soc, 22, 20, 1920, establishes independently 

 the formulae of the problem of cyclical quartisection, first given 

 (but not passed) by V. A. Le Besgne in i860, so far as they are 

 necessary for the problem of octosection. 



W. F. Sheppard in a memoir entitled " Reduction of Error 

 by Linear Compounding " proves certain general theorems by 

 means of which the analysis required in two problems in the 

 theory of error can be considerably shortened. These problems 

 are as follows. The data are a set of quantities, m<, uxu^ , . ., 

 which are regarded as representing certain true values, Uo Mi 

 U2 . . ., with errors eg e\ e^ . . ., so that u^ = u^ + e^. In the 

 first problem it is assumed that the sequence of w's is fairly 

 regular. In the second u^ is a polynomial in r of degree j, 

 and the problem is to find the coefficients in this polynomial 

 by the method of least squares. The general theorems referred 

 to allow other problems in which the assumptions are not so 

 narrow as in these two problems to be dealt with. 



In a paper entitled " On a Parabolic Equation of the rth 

 Degree for any Graphically Faired Curve " {Phil. Mag., vol. xl., 

 No. 238) T. C. Tobin gives a convenient and rapid method of 

 obtaining the values of the constants a^ <2i . . . in the equation 



y = ag -{- aix + <22^' . . . + a,.x^ 



satisfied by a curve plotted with respect to a set of rectangular 

 axes. 



F. H. Safford {Bull. Amer. Math. Soc, 26, 1920) gives the 

 parametric equations of the path of a projectile when the air 

 resistance varies as the nth. power of the velocity. 



In Acta Math., 42, 1920, David Hilbert publishes his address 

 to the Academy of Sciences of Gottingen delivered in 191 7 and 

 L. P. Eisenhart his address to the joint meeting of the American 

 Mathematical Society and the Mathematical Association of 



