98 



NATURE 



[July 21, 1923 



There are two positive roots, both between o and i . 

 The greater is the more suitable for our series. It is 

 very nearly equal to unity and it was found to be 

 very nearly 32/33. 



As a second approximation n \ is then equal to 



<«+(l/a)\ "*■'•/»'/ t\ '«-="/=» 



V2.(^-+(^/^)) "*■'""' (« + ;)' 



(5) 



r i/6+(l-a)/an 

 L'"^ 2{«+(l/a)} J' 

 where 1/0 = 32/33. 



From this expression, which is affected by an error 

 of order - i/^6on^, 10! was calculated: 



10 1 = 362,8806 (an error of 6). 



The approximation (4) gives 



101 = 362,8787 (an error of 13). 



From the original value of a, the second approxi- 

 mation will be 



e 



:oo8o,i875|_ (^) 



The error in this case will be less than 1/2000 xw^ 

 of the whole. 

 (6) gives 



101 = 362,8801 (an error of i). 



Forsyth's approximation (4) has an error of order 

 i/240«^. It will be seen that the first approximation 

 (3) is a remarkably good one, and the expression is 

 quite good for calculation purposes. The value of n ! 

 may be calculated in a very short time. 



Mr. Soper's expansion, taken to the same order 

 as (4), gives 



10 1 = 362,8792 (an error of 8), 



with an error of order i^oon^. The second approxi- 

 mation (6) derived in the same way as our first 

 approximation is exceedingly accurate, and is better 

 than that of (4) ; it is also better than Mr. Soper's, 

 which in turn is better than Prof. Forsyth's (4). 



Prof. K. Pearson has given in Biometrika, vol. vi., 

 a very close approximation to the value of n ! This 

 takes account of terms up to i/w* and partially of the 

 term in i jn^ : 



^°g ~S;^ "= °"399o899 -f J log w 



, „ . 25°-623 



+ -080,929 sm — — ~^. 



7t 



(7) 



On evaluating 10 ! by means of this expression, 

 it is found that the exact value is given to the nearest 

 unit. 



My chief aim in this note has been to show that a 

 very good first approximation may be obtained 

 without the use of any terms of the exponential and 

 that the resulting expression is useful for computing 

 factorials. 



It may be of interest to give the values of i ! 2 ! 

 and 10 ! found from these approximations in a single 

 table : 



Biometric Laboratory, 

 University College, London. 



NO. 2803, VOL. I 12] 



James Henderson. 



Dr. Kammerer's Alytes. 



May I reply in a few words to Dr. Bateson's brief 

 letter on Kammerer's Alytes, which appeared in 

 Nature of June 30 ? 



Dr. Bateson states that when the nuptial callosities 

 of genera allied to Alytes are described as appearing 

 on the " inner " sides of the fingers, the word 

 " inner " means the radial side and not the palmar 

 surface. 



This is quite true, but the callosity on the radial 

 edge of the finger involves the palmar surface also, 

 as Dr. Bateson may convince himself by insp>ecting 

 Boulenger's figures, and as, indeed, is demonstrated 

 to every student when he is shown the nuptial 

 callosity of the male Rana. 



Further, I learn from a letter from Dr. Kammerer 

 that in the specimen of Alytes shown at the Linnean 

 Society, the callosities extend round the radial edges 

 of the fingers on to the dorsal surface, and that he 

 would have demonstrated this to any one who had 

 raised this point while he was explaining his specimens 

 before the meeting. 



Readers of Nature are thus now in a position to 

 judge what ground there was for Dr. Bateson's 

 objections. E. W. MacBride. 



Imperial College of Science, 

 South Kensington, 



London, S.W.7, Jul}' 4. 



Molecular Interruption. 



In reply to Mr. R. d'E. Atkinson's criticism 

 (Nature, March 10, p. 326) of my note on the possi- 

 bility of selective molecular interruption, I should 

 like to point out that so far from attempting to 

 dispose of the validity of the ordinary treatment 

 and claim the effect in question for " infinite free 

 path," I had already shown (Nature, July 22, 

 vol. no, p. 112) the reverse to be the case, and that 

 such an effect is not then possible. 



It is manifestly clear that it is illogical to conclude, 

 however, because this is the case with " infinite 

 free path " {i.e. in the absence of intermolecular 

 collision in the system), that it must also be true 

 for a system in which intermolecular collisions exist, 

 with long free paths relative to the diameter of the 

 directing vessel employed, the particular and special 

 case alone dealt with in my note. 



Mr. Atkinson's misinterpretation appears to have 

 arisen from his overlooking my words " molecules 

 issuing from collision in circle O," since his statement 

 " all points on their long paths may equally be 

 taken " as being in O is otherwise unintelligible. 



His statement that I have admitted the length of 

 the free path to be irrelevant is not correct. The 

 excessive downward bias to which he refers is, in 

 my opinion, due entirely to the fact that molecules 

 proceeding from collisions (with equal probability of 

 motion in all directions) are interrupted by the vessel 

 before the end of their normal free path period, 

 when they are moving in certain specific directions ; 

 and are uninterrupted throughout the whole of their 

 normal flight, when they are moving in other specific 

 directions : a selective redirection or elimination 

 of the former class which must continuously be 

 leaving a corresponding preponderance of the latter — 

 a conclusion which more careful calculation confirms. 



Iving's College, 



University of London, 



Strand, W.C.2. 



Arthur Fairbourne. 



