October 25, 19 17] 



NATURE 



55 



through which the less luminous surface layer of the 

 interior gases becomes visible. The varying frequency 

 of spots is accounted for by supposing that at mini- 

 mum the heat of the central nucleus is prevented from 

 escaping by a photosphere of relatively great thickness, 

 and that afterwards, owing to contraction, the tem- 

 perature of the nucleus increases to such an extent that 

 the photosphere becomes attenuated and subject to 

 perforations in the form of spots and pores. Radiation 

 from the nucleus is then facilitated, so that the photo- 

 sphere again increases in depth, and eventually pro- 

 duces another minimum. The chromosphere, promin- 

 ences, and corona are regarded by Dr. Brester as 

 effects of a permanent aurora, which is maintained by 

 electrons projected from the photosphere. 



THE NEW PHYSICS. 



COPIES have reached us of five of Prof. Levi- 

 Civita's recent mathematical papers,' three of 

 which deal directly with Einstein's theory of gravi- 

 tation, and suggest some remarks on the aspect of 

 theoretical dynamics, as it appears at present to a 

 comparative layman unable to criticise rival theories 

 in detail. Speaking broadly, we may say that the 

 theory of mathematical physics is based upon a com- 

 paratively small number of fundamental differential 

 equations. Until recently time was explicitly or im- 

 plicitly treated as the independent variable, in terms 

 of which the other variables had to be found ; and 

 all phenomena were supposed to take place in a three- 

 dimensional Euclidian space, where we can use the 

 formula ds^ = dx- + dy^ + dz^ for the distance between 

 two very near points. In the theory expounded by 

 Minkowski and others we have a different formula, 

 ds- = c^df — {dx' + dy^ + dz^), where we may regard dt 

 as an element of time, and speak of a "world-point" 

 {x, y, z, t) determined not only by its position, but 

 also by its age. Einstein has developed his gravita- 

 tion-theory from the general expression, 'IgfjdXidXj 

 {i,j — o, 1,2,2), assumed for ds', where ds is an ele- 

 ment of distance in a four-dimensional space. (It 

 may be remarked that in the previous theory, as 

 Minkowski pointed out, we might take dt as a varia- 

 tion of a co-ordinate distance ; then phenomenal pro- 

 cesses in our space might be regarded as "sections," 

 so to speak, of a four-dimensional system.) 



With Einstein's form of ds^ we can at once use all 

 the known geometrical theory of infinitesimal geo- 

 metry in four dimensions, and, in fact, the well-known 

 symbols of Riemann and Christoffel directly enter 

 into Einstein's gravitation formulae. This is a matter 

 of mathematics merely; the most striking fact, from 

 the physical point of view, is that Einstein has used 

 his formulae successfully to account for the secular 

 motion of the perihelion of Mercurv. This does not 

 show that Einstein has said the last word on the 

 theory of gravitation, but it does show that these 

 post-Newtonian theories provide a calculus which gives 

 a better image of actual facts than the purely New- 

 tonian theory seems able to do. The more predictions 

 the new theorj' can give us, which are verified by 

 experiment, the mote we shall be inclined to trus't 

 it; and this is quite independent of what we call the 

 "real meaning" of the symbols involved. For in- 

 stance, Prof. Levi-Civita's paper No. 2 seems to 

 show that if we could produce a sufficiently strong 

 magnetic field, we should find it inducing upon the 

 three-dimensional space to which, so far, our intuition 



/ ^ 5<'2 ''S'a'ics Einsteiniana" ; (2) " Health fi.ica di alciini spazi . . . "l 

 yl <.xt" • e^Pressione analitica snettante al tensore gravita/i'onale . . . "; 

 U) Nozione di piralletismo in una varielh nualunqiie . . ."(■;) " Siille 

 v"/? i^"T'' v""^"- '"K''w*?*='-'.' il^- (''• ""^ (3) ^^^ reprints from Ren^ic. 

 1^ f J-/^', "'fL^,'^" ^■"'"K '?°""'' '9'7>; (i)from Re„>iic. del Circ. 

 Mat. ^//'«/^r«w (Palermo, 1917); (5)from Attie Memoric delta R. Accad. 

 dt Padoz'a (Padua, 19 17). 



NO. 2504, VOL. 100] 



appears to be confined, a corresponding "curvature" 

 measured by i/R*, where R is a length. Assuming 

 that the field is one of 25,000 gauss, the author de- 

 duces that R = 3 . 10" cm., or about ten million 

 times the mean distance of the earth from the sun. 

 As he points out, there is little hope of testing this by 

 experiment, but he obtains a formula for the velocity 

 of light, V = c, exp(x/R) + C2exp(-x/R), with a damp- 

 ing coefficient in the second term, which he suggests 

 might come within the range of observation. 



Philosophically, the trouble still seems to be about 

 time, in the philosophical sense. If we could look 

 at the universe sub specie aeternitatis, we might per- 

 haps find our greatest delight in its unchangeable 

 perfection ; but so long as we are constrained by pro- 

 cesses (even processes of thought), time, in some- 

 sense or other, is apparently indispensable, and if we 

 evict it from one habitation, we may expect it forth- 

 with to be in occupation of another. G. B. M. 



METEOR ORBITS. 



A PAMPHLET on "The Determination of 

 -^"^ Meteor Orbits in the Solar System," by G. von 

 Niessl, has just been published in Smithsonian Miscel- 

 laneous Collections (vol. Ixvi., No. 16, Washington, 

 1917). The pamphlet is a translation by the late Cleve- 

 land Abbe of a paper published in the " Encyclopadie 

 der mathematischen VVissenschaften," dated Vienna, 

 1907. The author, who has had considerable experi- 

 ence in computing meteor paths and orbits, gives his 

 views as to the mathematical treatment of the subject. 

 He indicates the best method to be followed in deter- 

 mining the radiant and geocentric velocity of meteors 

 and fireballs of which multiple observations have been 

 obtained. Not the least interesting part of his discus- 

 sion is that in which he deduces the mean errors in 

 the results : — 



Mean error of azimuth = 5-8", 351 observations. 



Mean error of apparent altitude = 41°, 235 observa- 

 tions. 



Mean error of radiants = 33°, 43 cases, 537 observa- 

 tions. 



Mean error of inclination = 65°, 250 observations. 



The radiant positions of the chief periodical showers 

 he gives to within 1° of probable error. 



Tables are furnished of the average terminal velocity 

 and altitude of meteors, from which he concludes that 

 they "can penetrate deeper into the atmosphere in 

 proportion as they move with a low velocity " — a fact 

 previously well ascertained. With regard to atmo- 

 spheric resistance, von Niessl's opinion is that direct 

 observations make it probable that the velocity of 

 meteors in the upper atmospheric regions is slighter, 

 while in the lower strata of the air it is greater, than 

 theoretical views. 



The masses of fireballs and shooting-stars are dis- 

 cussed from various data. Prof. A. S. Herschel dealt 

 with this part of the subject many years ago, and 

 held the view that a first magnitude' meteor is usually 

 a few grams in weight, while the very small meteor's 

 are only the fraction of a gram. V. F. Sands found 

 from the Leonids of 1867 that the average mass, or 

 weight, of a meteor equal to Jupiter in brightness was 

 067 gram, while a fourth magnitude object was only 

 0006 gram. 



Von Niessl finds it necessary to assume decidedly 

 hyperbolic orbits for the majority of meteors, for thei'r 

 "observed geocentric velocity far exceeds the limits for 

 parabolic orbits. Therefore the large meteors in 

 general are undoubtedly of interstellar origin." Schia- 

 parelli arrived at similar conclusions half a century 

 ago. 



The paper is an instructive contribution to the litera- 



