December i, 1921] 



NATURE 



435 



pointed out by the present writer (Bulletin of the 

 American Mathematical Society, vol. 20, 1914, pp. 

 524-31), but the exceptions there are all imaginary, 

 for the slope of the curve at the initial point has to 

 be equal to *J — \. The result is that the limit then 

 takes a discrete set of values. It has to be one of 

 the numbers 094, o-86, o-8o, 074, . . . ; the par- 

 ticular one depending on the contact of the curve with 

 the minimal straight line at the given point. If there 

 is no contact the value is unit>-, but if the contact is 



of order fe-i, then the value is ~. 



For Euclidean space of three dimensions it comes 

 out that the limit can take all real and complex values. 

 The Minkowski space is four-dimensional, and here 

 also continuous variation is possible ; but the essential 

 point is that on account of the minus signs in the 

 interval formula the exceptional cases which were 

 imaginary' in the Euclidean geometry- now become 

 physically real. This does not mean, however, that 

 experimental verification would be easy. Particles 

 have been found in radio-emanations where the initial 

 velocity is more than nine-tenths that of light, but as 

 Icmg as the velocit\' is actually less than that of light 

 the limit we are dealing with is unit}-. As the 

 velocity' is increased the limit thus remains unity. It 

 takes the exceptional value 094 only when the initial 

 velocity is actually that of light. Therefore, as the 

 initial speed is increased continuously up to c, the 

 limit jumps suddenly from unity to 094. The limit of 

 the limit equals i, but the actual value attained equals 

 094. Such a discontinuity is perhaps heyond the 

 possibility of experimentation, hut there is no douht 

 of its theoretical validity. 



If, instead of the Minkowski formula, we use the 

 general Einstein relativity theory, we have a curved 

 manifold of four dimensions instead of a flat mani- 

 fold. The formulae, involving the potentials ga, are 

 much more complicated, but again we find excep- 

 tional values for the limit of the ratio of the arc to 

 the chord whenever the initial velocit\- is that of light 

 — that is, whenever the world-line is tangent to a 

 null geodesic on the cur\'ed manifold representing the 

 field of gravitation. Edward Kasner. 



Columbia University*, New York, September 20. 



The Softening of Secondary X-rays. 



Dr. a. H. Compton's letter to Nature of Novem- 

 ber 17, p. 366, on the softening of secondary X-rays 

 directs further attention to a problem of very great 

 importance. There is distinct evidence with these 

 rays of a change of periodicity which varies with the 

 angle of scattering. Such a variation is, perhaps, 

 unique in physics, and no satisfactory explanation of 

 the facts has been found. Let us consider the his- 

 torv of the case. 



In 1913 (Phil. Mag., vol. 26, p. 611) I stated that 

 when homogeneous rays struck any target the scat- 

 tered rays were softer (i.e. of lower frequency), and 

 tiiat this softening increased with the angle of scat- 

 tering. This view was a deduction from experiments 

 with a heterogeneous primary beam consisting of 

 the 7-rays of radium. The experimental results were 

 verified by Dr. Florance (Phil. Mag., vol. 27, p. 225, 

 1914). 



In 1919. working at University College with hetero- 

 geneous X-rays, I again found that the secondary- 

 rays were less penetrating than the primary. At the 

 time I was not successful in obtaining a homo- 

 geneous primary- beam of sufficient intensity', for such 



NO. 2718, VOL. 108] 



a beam can be obtained only by reflection from a 

 crystal. Later on I was informed that Mr. S. J. 

 Plimpton, on continuing the problem at University 

 College, had found evidence which apparently showed 

 that when the primary rays were homogeneous, the 

 secondary rays were of the same frequency. Hence, 

 in a paper to the Journal of the Franklin Institute 

 (November, 1920), I endeavoured to account for the 

 softening observed with heterogeneous X-rays by 

 assuming them to consist of thin pulses, which 

 became thicker and softer, and hence of smaller 

 apparent frequency as the scattering angle increased. 

 Plimpton published his results in the Philosophical 

 Magazine for September, 192 1. 



The work of Compton (Phil. Mag., vol. 41, p. 749, 

 192 1, and Phys. Rev., vol. 18, p. 96, 192 1), however, 

 confirms my original view, although it may perhaps 

 be advisable to substitute the term " secondary* rays " 

 for "scattered rays." On working with homogeneous 

 X-ray beams he finds the same change as when 

 ordinary X-rays of corresponding penetrating power 

 are used. Thus secondary X- or 7-rays, even when 

 homogeneous, decrease in frequency as the angle of 

 scattering increases, and this remarkable relation is 

 independent of the scattering medium. 



I have always looked on the secondan,- rays as 

 scattered rays, because the theory of scattering first 

 given by Sir Joseph Thomson ("Conduction of Elec- 

 tricitv through Gases," 1906, p. 321), and since 

 developed bv other writers, accounts so well for the 

 variation in intensit\' of the secondary radiation with 

 angle of scattering, and also for the obser\'ed polarisa- 

 tion of the secondary radiation. This theor}-. how- 

 ever, in its present form does not account for the 

 changes in periodicity referred to above. 



Compton suggests' that the greater part of the 

 secondan*- radiation is fluorescent, i.e. that it is pro- 

 duced by the secondan,' /3-ravs which are always 

 emitted when X- or -y-rays strike anv substance, and 

 that the change in periodicity can be accounted for 

 bv means of the Doppler principle. I believe that it 

 can be proved that only a negligible portion of the 

 secondar\- X-rays can be accounted for in this way, 

 and hence that' this suggestion does not help us out 

 of our diflRculties. J. A. Gray. 



McGill L'niversitv, Montreal, November 12. 



University Relief for Central Europe and Russia. 



T SHALL be grateful for space to bring before 

 readers of Nature the following facts concerning the 

 activities of the Imperial War Relief Fund, Universi- 

 ties' Committee. This committee, which was created 

 at an Inter-University Conference which met at Uni- 

 versity College, London, on July 7, 1920, at the 

 invitation of Lord Robert Cecil, and under the 

 auspices of the Imperial War Relief Fund, has set 

 before it the aim of presenting to the British uni- 

 versities the appeal of the universities in the war- 

 stricken areas of Europe. 



During the first year of the existence of the Uni- 

 versities' Committee 32,000/. was raised in co-opera- 

 tion with ever\- university- in Great Britain and 

 Ireland. The committee at the opening of this uni- 

 versitv year carefully considered the problem of the 

 Central European universities at the present time, 

 and decided that it would be absolutely necessary- for 

 us to maintain the relief work promoted bv the com- 

 mittee in co-operation with universities all over the 

 world throughout the coming year. 



I mav sav briefly that the financial panic which 

 has swept through Austria in particular during the 



