790 



NATURE 



[February 17, 1921 



placements, and to investigations of the character- 

 istic behaviour of spectrum lines, as all such data 

 will have a part in solving one of the most absorb- 

 ing questions in cosmic physics. 



Evershed adduces his observations upon the 

 spectrum of Venus as evidence of an " earth- 

 effect " driving the gases from the earth-facing 

 hemisphere of the sun, and he would by this hypo- 

 thetical action explain the observed displacements 

 of the solar lines, and thus negative the deduction 

 from the Einstein theory. Two series of Venus 

 observations have been made by Dr. S. B. Nichol- 

 son and myself. The details will appear in a 

 forthcoming Contribution from the Mount 

 W^ilson Observatory. Our observations indicate 

 that the displacements of the Venus lines to the 

 violet relative to skylight are due to non-uniform 

 illumination of the slit when the guiding is done 

 upon the visual image, the effect increasing with 

 the refraction and becoming more evident "the 

 smaller the image. The explanation is based upon 

 the observation that spectrograms taken at low 

 altitudes give larger displacements to the violet 

 than those taken on the same night at higher alti- 

 tudes, and that the displacements correlate with 

 the cotangent of the altitude and the reciprocal 

 of the diameter of the planet at the time of 

 observation. 



In respect to the observations at Mount Wilson 



on the lines of the cyanogen band at A3883, I have 

 as yet found no grounds for considering them seri- 

 ously in error. The explanation of the results 

 adverse to the theory based upon dissymmetry 

 appears inadequate (Observatory, p. 260, July, 

 1920), and the assumption that the adverse results 

 are due to superposed metallic lines seems to be 

 negatived by the observations of Adams, Grebe, 

 Bachem, and myself that for these lines there is 

 no displacement between the centre and limb of 

 the sun. Metallic lines as a class shift to the red 

 in passing from the centre to the limb. If, then, 

 metallic lines are superposed on these band lines 

 in such a way as to mask the gravitational dis- 

 placement to the red when observed at the centre 

 of the sun, this should be revealed by a shift 

 to the red at the limb. 



The lines of the cyanogen bands are under in- 

 vestigation in the observatory laboratory both as 

 reversed in the furnace and as produced in the arc 

 under varying pressure. The measures show no 

 evidence of a displacement to the red under de- 

 creased pressure as indicated by Perot's observ'a- 

 tions. 



The present programme at Mount Wilson aims 

 at an accumulation of varied and extensive data 

 that will furnish a suitable basis from which to 

 approach the general question of the behaviour of 

 Fraunhofer lines relative to terrestrial sources. 



Non-Euclidean Geometries. 



Bv Prof. G. B. Mathews, F.R.S. 



THE ordinary theory of analytical geometry 

 may be extended by analogy as follows : 

 Let x-^, X.2, . . . Xn be independent variables, each 

 ranging over the complete real (or ordinary com- 

 plex) continuum. Any particular set (%i, x^, ■ ■ ■ Xn), 

 in that order, is said to be a point, the co-ordinates 

 of which are these Xi ; and the aggregate of these 

 points is said to form a point-space of n dimen- 

 sions (Pn). Taking r<iri^ a set of r equations 

 <^j=o, (l>2 = o, . . . (f>r = o, connecting the co-ordin- 

 ates, will in general define a space P„-r contained 

 in Pn. Theorems about loci, contact, envelopes, 

 and the principle of duality all hold good for this 

 enlarged domain, and we also have a system of 

 projective geometry analogous to the ordinary one. 



Physicists are predominantly interested in 

 metrical geometry. The ordinary metrical formulae 

 for a P3 may be extended by analogy to a Pn ; 

 there is no logical difficulty, but there is, of course, 

 the psychological fact that our experience (so far) 

 does not enable us to " visualise " a set of rect- 

 angular axes for a Pn if n>>3 ; not, at least, in 

 any way obviously analogous to the cases n — 2, 3. 



In ordinary geometry, for a P3 we have the 

 formula 



ds^ = d-Tj' -I- dxo^ + dx^ 



for the linear element called the distance between 

 two points [x), (x + dx). Riemann asked himself 

 the question whether, for every P„, this was neces- 

 NO. 2677, VOL. 106] 



sarily a typical formula for ds, on the assumption 

 that solid bodies can be moved about in space 

 without distortion of any kind. His result is that 

 we may take as the typical form, referred to ortho- 

 gonal axes, 



where N = i -h \d2,x-, 



and o is an arbitrary constant, called the curva- 

 ture of the P„ in question. This curvature is an 

 intrinsic property of the P„, and should not be 

 considered as a warp or strain of any kind. When 

 a = o, we have the Euclidean case. .'\s an illustra- 

 tion of the theory that can be actually realised, 

 take the sphere x^-\-y^-\-z^ = r'^ in the ordinary 

 Euclidean P3. By putting 



D.^, Dy, Dz = 4r2M, i\.rh.\ {u^ + v^ — ^r'^)r, 



the equation x'^ + y^ + s^ = r^ becomes an identity, 

 and we may regard the surface of the sphere as a 

 Po with [u, v) as co-ordinates. The reader will 

 easily verify that 



ds'' = {du^ + dv^)^{i+~ (u2 + 1^2)2 ; 

 4r 



so we have a case of Riemann 's formula with 

 a = r-^. We cannot find a similar formula for 

 the surface of an ellipsoid, because a lamina that 

 "fits" a certain part of the ellipsoid cannot be 



