August 25, 1923] 



NA TURE 



283 



hour it will be advantageous to take them in opposite 

 directions {e.g. south-west and north-east). A reseau 

 of five photographs would practically cover the whole 

 visible sky when an average lens is employed, and it 

 is accordingly recommended that, when possible, one 

 photograph should be taken towards each of the 

 points north, east, south, and west, and one towards 

 the zenith. Photographers should be particularly 

 careful to mark their plates in some way, so that the 

 photographs in the different directions may be readily 

 recognised after development ; the inclusion of a small 

 strip of horizon might be advisable for this purpose. 

 In the case of the zenith photograph, a small part of 

 some object might be included [e.g. the top of a tree 

 or the corner of the roof of a house) to indicate the 

 orientation of the plate. 



The main object is not to secure artistic effects, 

 but rather to obtain clearly defined records of the 

 cloud forms present, and therefore " contrasty " re- 

 sults are preferable. 



Photographers who are willing to take part volun- 

 tarily in this work are invited to send their names to 

 one of us at Stoner Hill, Petersfield, and these volun- 

 teers will be supplied with the necessary instructions 

 when these are ready for distribution. At the request 

 of Col. Delcambre, of the French Meteorological 

 Service, instructions for taking the photographs have 

 been drawn up by one of us and are to be circulated 

 internationally. C. J. P. Cave. 



G. AuBOURNE Clarke. 



An Einstein Paradox : an Apology. 



Allow me to express regret for having misinter- 

 preted Prof. Einstein's symbols. My mistake was 

 caused by the fixed idea that it was impossible for 

 Ki in motion to learn anything about the signal at L 

 until the light reached him. 



I owe to Mr. C. O. Bartrum the explanation that 

 there are three events, namely, (i) the emission of 

 light-signal at L ; (2) its reception by Ki ; (3) its 

 reception by K ; and that each requires its own 

 double set of space-time co-ordinates; thus {x-^^, t^, 

 {^2. ^2). (^3, ^3) in K's system and the same letters with 

 accents for Kj's. There will then be three pairs of 

 Einstein equations. 



I find, however, from letters received, that opinions 

 differ as to the interpretation of the ^'s. Some think 

 that they are the actual times recorded by the clocks ; 

 others that they have to be corrected by allowances 

 for the passage of light. Some think that a body in 

 motion actually contracts and that a carried clock 

 goes slow ; others that the body only seems to 

 contract and that each of the two observers thinks 

 that the other's clock goes slow. The latter have a 

 difficulty in explaining the constant c. 



The simple problem of which the Newtonian solu- 

 tion was given in Nature of June 2 ought to admit 

 of a solution by relativity methods. I should be 

 greatly obliged to any of your readers who would 

 send me one showing the time on K's clock when the 

 signal reaches K, viz. Xi/v + xjc. R. W. Genese. 

 40 London Road, 



Southborough, Kent. 



Colour Vision and Colour Vision Theories. 



Prof. Peddie, in Nature of August 4, p. 163, has 

 dealt with some of my strictures of the trichromatic 

 theory. Whilst nothing can be said against his 

 mathematical presentation of the theory, it can easily 

 be shown that, when a case of colour blindness is 

 fully and carefully examined, the mathematical 



NO. 2808, VOL. I 12] 



presentation will not account for the facts. All the 

 facts which are explained by the trichromatic theory 

 are, however, consistent with my theory. 



The trichromatic theory becomes more and more 

 complicated with subsidiary hypotheses, inconsistent 

 with each other. I have examined a man stated to be 

 completely red blind, but tested with my lantern he 

 recognised red as easily as a normal-sighted person. 

 How do 50 per cent, of the dangerously colour blind 

 get through the wool test ? The trichromatic theory 

 completely fails to explain the trichromic class of 

 the colour blind. The trichromic have no yellow 

 sensation, regarding this region of the spectrum as 

 red-green and marking out in the spectrum a mono- 

 chromatic division including yellow, orange-yellow, 

 and yellow-green. 



If the trichromatic theory were true the point 

 where the hypothetical curves cut should be shifted 

 towards the defective sensation ; this is not found. 

 Let the trichrome now be examined by colour-mixing 

 methods, and he may make an equation R + G -h V = W, 

 with too much red in the mixed light, and then make 

 an equation with too much green in the mixed light. 

 Again, he may agree with the normal match, or in 

 other cases only agree with the normal match when 

 the comparison white light is diminished in one case 

 or increased in another, thus matching two white 

 lights of different luminosities. 



F. W. Edridge-Green. 



London, August 7. 



Stirling's Theorem. 



The recent correspondence in the columns of 

 Nature on this subject prompts me to add to the 

 collection a formula which I deduced about three years 

 ago. It was then communicated to a mathematical 

 friend, but has not otherwise been published. 



The ordinary Euler-Maclaurin series for logs w ! is 



log ^/2■^■ +{n + ^) log n -n 



-f I/I2M - i/36o»'+ i/i26on* . . . 



It is easily shown that the last three terms printed 

 above are reproduced exactly by the first three terms of 

 the binomial 



I2n\^ 2ion^J 



-7/H3 



while the simpler binomial 



i2nV'^i'iny ' '^^ i2«\i'5w''+8/ ' 



reproduces exactly the terms in i/w and i/w* and 

 very approximately the term in i/m'. Adopting the 

 simpler form, we have 



log n ! = \og\/2ir+ [n + ^) log n - w -I- : 



[5«*-l-8>' 



or passing to common logs (M = 

 logiow! = 0-39908993 . . . 



: modulus). 



+ (n + ^) logio w-wM-^Y 



M 



2«\i5«*-f- 



1/16 



This formula gives for i ! (true value i), i -00007 • • • ; 

 for 2 ! (true value 2), 2-000002 . . . ; for 3 ! and 5 ! no 

 discrepancy is shown by 7-figure logs and 9-figure logs 

 respectively. The degree of approximation is there- 

 fore high and even remarkable ; but it may be 

 doubted whether this formula or any of those under 

 discussion is really to be preferred to the direct use 

 of the series of which we can easily take as many 

 terms as may be required for the order of accuracy 

 desired. ' G. J. Lidstone. 



9 St. Andrew Square, Edinburgh, 

 July 24. 



