November iS, 1922] 



NA TURE 



669 



If the traveller happened to be a relativist, his faith 

 in the transformation formulae would receive a rude 

 shock when, instead of the seven and a half days 

 he had calculated, he found on returning that the 

 earth had aged no less than two hundred years. 



There is also an obvious slip as regards the speed 

 necessary to produce so large a difference in computed 

 time (assuming the paradox to stand good). An 

 observer travelling out and back with a velocity of 

 one twenty-thousandth the velocity of light, or 

 ■9-3 miles per second, would only expect a difference 

 of one-twelfth of a second in either direction between 

 his own time and earth time after two years. 1 This 

 is perhaps fortunate for us, as the earth travels with 

 twice this velocity, or 18-5 miles per second in its 

 orbital course. The cannon-ball would indeed have 

 to be projected with a velocity of within one twenty- 

 thousandth of thevelocityof light,«.e.,y = c(i - 1/20,000) 

 or with the incredible speed of about 185,990 miles 

 per second, to produce the result stated. 2 



This, however, which is plainly only a lapsus 

 calami, is of small importance. The difficulty is 

 created, not by the magnitude of the paradox, but 

 by its existence, and the contradiction it opposes to 

 common sense. If true, it throws the whole relativity 

 doctrine into the lap of metaphysics, from which, 

 if we are to believe M. Nordmann, Einstein was 

 determined to rescue it. " La theorie d'Einstein est 

 nee de problemes poses par V experience. Elle est nee 

 des faits, et son auteur insiste avec beaucoup de 

 vigueur sur ce point. . . . Elle est tout le contraire 

 d'un systeme metaphysique " (p. 134, loc. cit.). 

 Obviously this paradox, in any of its forms, can never 

 be subjected to the test of experiment ; and as it 

 is a fundamental principle with Einstein that nothing 

 must enter into his theory (and therefore that nothing 

 must interfere with his theory) that cannot be so 

 tested, is not the difficulty "thereby automatically 

 ruled out of consideration ? These are deep waters, 

 into which a sciolist like myself has to venture 

 carefully, even when it is done of necessity, by way 

 of question, in search of information from competent 

 authority. H. C. Browne. 



Dublin, October 26. 



There is, as Mr. Browne points out, a lapsus 

 calami in my ([notation. The supposed velocity of 

 the cannon-ball is, not a twenty-thousandth of, but 

 less by about a twenty-thousandth than, the velocity 

 of light. It is an often-quoted paradox, which I 

 heard for the first time from M. Langevin in his 

 address to the Philosophical Congress of Bologna in 

 1911, and the discussion of it occupies a large part 

 of M. Bergson's book. With regard to the paradox 

 itself, it is, as Mr. Browne very well points out, not 

 a paradox for the relativist but an illustration of 

 the consequence of rejecting the principle of relativity. 

 In exactly the same way Zeno's paradoxes were not 

 paradoxes for Zeno but arguments for his doctrine 

 that nothing moves. The principle of relativity is 

 that it is possible to pass to a completely different 

 frame of reference without breach of continuity, 

 provided that the space-time coefficients vary to 

 maintain the ratio constant. The paradox shows 

 the form which the breach of continuity will assume 

 if with common-sense we suppose the change of the 



Taking e=i, and v = i/: 

 (=CH-\/: 



■when t' - 

 second. 



/400. 



■2 years-f- 



3 c=r, u=i— 1/20,000. 

 Therefore t = ioof . 



000= f (1 + 1/800,000,000+a negligible), 

 63,072,000 

 3oo 000 000 sec0D " s > or ' — * is less tnan l l l 



t,2 = T7>7^n _a negligible, or Jz-v'=— 



system of reference not to be compensated by a 

 variation in the space-time co-ordinates. There are, 

 in fact, two alternatives. I may conceive my traveller 

 retaining the dimensions of his old system in his 

 new system, then he will become a kind of ephemeral 

 insect or microbe in his new environment, for his 

 proportions will be incommensurate with his old 

 proportions ; or, I may conceive him automatically 

 shrinking or expanding in his dimensions proportion- 

 ately to the change in his environment, then, however 

 much the system changes, he can never become 

 aware of it. This is what I referred to in my article 

 as the relativity of magnitudes. The paradox dis- 

 appears in the principle of relativity ; it arises because 

 common-sense is accustomed to the view that space 

 and time are constant and invariable. 



H. Wildon Carr. 

 November 1. 



Waterspouts. 



Corroborating the letter of Dr. G. D. Hale 

 Carpenter in Nature of September 23, p. 414, refer- 

 ence may be made to a note in Monthly Weather 

 Review, 43, p. 550-551, 1915, where a funnel or 

 pendant seen near Cape San Lucas, Lower California, 

 is described and sketched ; the sheath or sleeve 

 seen by Dr. Hale Carpenter was very striking. The 

 phenomenon was under observation a considerable 

 time. 



Also, the following from my note-book on an 

 observation made in Manila, P.I. : 



" igig V. 24 d. 6 h. ij nt. p.m. — Under a thunder- 

 storm developing in N., from my window I saw a 

 small tornado funnel stretching downward in N.W., 

 obliquely toward W. or S.W. It did not reach half- 

 way to earth ; the sun was so low that a flood of 

 yellow light poured horizontally under the cloud, 

 and the funnel was brilliantly lighted. The figure 

 and description given in my note, Monthly Weather 

 Review, November 1915, apply excellently, except 

 that the brighter illumination brought out the hollow 

 core better. The distance was greater, so that I 

 could not very well make out the lattice pattern." 



This one showed the sleeve or sheath very well. 

 Another, mentioned in the same note in the Monthly 

 Weather Review, a gauzy but large waterspout, 

 extending clear to the water, and causing there a 

 great powder-puff of spray, did not show the sleeve 

 at all. (This was near San Salvador, in the Bahamas ; 

 the position given by latitude and longitude is quite 

 wrong, inserted by some other hand.) 



Willard J. Fisher. 



Cambridge, Mass., October 16. 



no. 2768, vol. no] 



Tables of the Incomplete Gamma-Function. 



I should be greatly obliged if you could allow me 

 a little of your valuable space to state that Dr. J. F. 

 Tocher has kindly pointed out an error in my Intro- 

 duction to the above Tables. In a table on page xx 

 the wrong argument has been inserted to the correct 

 value of the function. 



An errata slip has now been issued, and will be 

 inserted in all future volumes sold. This slip will be 

 supplied by the Sales Office, H.M. Stationery Office, 

 Princes Street, Westminster, to all past purchasers of 

 the work, and is arranged so that owners of the Tables 

 can paste them over the offending matter. 



I can only apologise sincerely to purchasers of the 

 book for this inadvertency. Karl Pearson. 



Department of Applied Statistics, 

 University of London, University College.W.C.i. 



X 2 



