164 



NATURE 



[April 26, 19 17 



A Sylow Factor TcCble of the First Twelve 

 Thousand Numbers, giving the Possible Number 

 of Sylow Sub-Groups of a Group of Given Order 

 between the Limits of o and 12,000. By H. W. 

 Stager. Pp. xii+120. (Washington: Carnegie 

 Institution of Washington, 1916.) Price 4.50 

 dollars. 

 The main object of this publication is to answer 

 the question : Given n, the order of a group, what 

 are the possible orders of such Sylow sub-groups 

 as it contains? This amdunts to finding all 

 divisors of n which are of the form p (kp + i), 

 where p is prime. For each 7i up to 11,999 the 

 table gives the complete resolution of n into its 

 prime factors, and the values of k (other than o and 

 2, which do not require entering) corresponding to 

 each prime factor. Each prime value of n is 

 entered in the body of the table in the form Pt] 

 for instance, the entry pgo? under 4639 shows that 

 the latter is the 627th prime in order of magni- 

 tude, taking pi=i. It is obvious that, apart 

 ■ from its special purpose, this table will be very 

 useful to arithmeticians ; every reasonable precau- 

 tion seems to have been taken to make it accurate, 

 and fortunately the table is of such a kind that 

 every single entry can be tested with very little 

 trouble, and any misprint almost certainly 

 detected, unless a number n has been entered as 

 prime, when really composite. Cases where 

 p [kp+i)=zn, and not merely a divisor of n, are 

 noted, such numbers are called Ps by the com- 

 piler — for instance, 1074 = 3 (3-i^9+i)> so 1074 

 is a P. On pp. xi and xii is a list of these num- 

 bers (1-12,229) in their natural order; and there 

 are interesting tables and graphs on the distribu- 

 tion of P numbers and primes. Supposing that 

 (f>{n) means the number of primes not exceeding 

 n, and \{/{n) the number of P riumbers not exceed- 

 ing n, the tables suggest that when n — >■ oo the 

 ratio ip{n)/4>(n) converges to a definite limit not 

 very different from e ; of course this is a mere 

 guess that might occur to anyone, but at any rate 

 to find a formula for >/'(n) analogous to Riemann's 

 for <f>(n) would be an interesting problem. It may 

 not be superfluous to add that the table does not 

 profess to enumerate actually existent Sylow sub- 

 groups for different values of n. 



LETTERS TO THE EDITOR. 



[The Editor does not hold, himself responsible for 

 opinions expressed by his correspondents. Neither 

 can he undertake to return, or to correspond with 

 the writers of, rejected manuscripts intended for 

 this or any other part of Nature. No notice is 

 taken of anonymous communications.] 



Aeroplanes and Atmospheric Gustiness. 



The question which Prof. McAdie raises in his letter 

 in Nature of April 12 is how to measure the un- 

 steadiness of the air as it affects an aeroplane. 

 Among those connected with flying the term "bumpi- 

 ness " is used to express the unsteadiness of the air 

 as it affects an aeroplane, and, in the absence of a 

 better word, this may be used here. The problem is, 

 then, to measure the "bumpiness" of the air by 

 meteorological means. 



As Prof. McAdie points out. gusts may occur in any 



direction, but gusts in different directions will not 

 equally affect an aeroplane, those in a vertical direc- 

 tion having the greatest effect. There is evidence to 

 show that the fluctuation in velocity of any individual 

 portion of air is, on the average, roughly the same in 

 any direction, so that, in view of the greater effect 

 of vertical gusts, the fluctuation of the wind, as given 

 by an anemometer, may give entirely erroneous in- 

 dications of the "bumpiness." The best example of 

 this is in the middle of a hot summer day, with a 

 light wind, and sky partly covered with small cumu- 

 lus clouds (themselves an indication of vertical cur- 

 rents). Under these conditions, the variation shown 

 by an anemometer may be less than three metres per 

 second, but the air will be very "bumpy" for an 

 aeroplane. 



On sunless days, with strong winds, the air is 

 "bumpy" on account of the eddy motion set up by 

 friction with the surface of the ground. If the con- 

 ditions be the same, the fluctuation is roughly pro- 

 portional to the mean velocity of the wind, but under 

 different conditions — e.g. between night and day — the 

 fluctuation may be very different for the same strength 

 of wind. When it is remembered that the mean 

 velocity of the wind does not in itself affect an aero- 

 plane when flying (except as regards getting from 

 place to place), it will be seen that the mean velocity 

 of the wind should not enter into measurements of 

 the "bumpiness" of the air. Further, the gustiness 

 of the air near the ground is of little importance, 

 except in getting off and landing, and also it cannot 

 be taken as an indication of the "bumpiness" of the 

 air at a height. 



To obtain satisfactory information, it would be 

 necessar)' to measure the fluctuation of velocity in 

 three directions and at various heights. Several 

 methods have been devised for obtaining the vertical 

 velocity, as well as the horizontal velocity and direc- 

 tion, near the ground; to obtain such data at a 

 heig^ht is very much more difficult. 



The "bumpiness" of the air might be measured 

 by an accelerometer carried on ah aeroplane ; but this 

 would not be satisfactory to the meteorologist, since 

 it would be difficult to discriminate between vertical 

 currents and horizontal gusts. It may be pointed out 

 that by the use of both a gravity-controlled and a 

 spring-controlled air-speed indicator, this is at least 

 theoretically possible. 



If we suppose a satisfactory means of measuring^ 

 the "bumpiness" to have been devised and standard- 

 ised, it would be possible to compare the average, 

 "bumpiness," say, at one place with that at another, 

 from which, no doubt, some useful information could 

 be obtained. But to make real advance in this part 

 of meteorology, it is necessary to go further and 

 endeavour to find the real structure of the atmosphere 

 and the causes which may give rise to this structure. 

 It is, therefore, to be hoped that the work of meteoro- 

 logists interested in this branch of the subject may be 

 devoted more and more to these fundamental problems 

 of cause and effect, rather than to the making and 

 tabulation of routine observations, other than those 

 made with the definite object of throwing light on 

 some particular problem. Gordon Dobson. 



Farnborough, April 16. 



In reply to Prof. Alexander McAdie 's request 

 (Nature, April 12) for a means of recording g-ustiness, 

 I venture to throw out the suggestion that this might 

 be done by observing what in German is called " wim. 

 mern." " Hoert Ihrs wimmern hoch vom Thurn ? Das 

 ist St^irm." This variation in the sounds heard frqm 

 church bells during gusty weather is due to the 

 irregular velocities in the atmosphere. That part of 



