July 15, 1909J 



NATURE 



69 



Musical Sands. 



May I record the existence of musical sands along the 

 shore at the Sandbanks, Poole Harbour? 



Some years ago the Poole authorities erected a series 

 of box groynes along this coast between Poole Head and 

 the Haven, and these have considerably increased the 

 natural accumulations of sand, so that it is " making " 

 everywhere, and the growth of the marram grass on the 

 dunes is in many places (independently of that recently 

 planted) rapidly extending seawards. 



The beach now, between each groyne, consists of wide 

 and fiat deposits of sand, shells^, and flint pebbles, but 

 about midway between the dunes and the sea, where the 

 sand is comparatively free from these, musical zones are 

 of frequent occurrence. 



In walking along the shore in a westerly direction, start- 

 ing from the first groyne, the sounding qualities of the 

 sand notably increase. Thus between the first and second 

 g'rovnes there are no musical patches, between the second 

 and third the sounds are very faint, and between each of 

 tne other groynes, until one reaches the last at the Haven 

 Point, the intensity of the sound increases. In a small 

 cove at the Point, formed by the last groyne (constructed 

 of barrels of concrete and an old ship), the sand is remark- 

 ably musical. 



The increase of sound observed when walking in a 

 westerly direction is due, I think, to the fact that the 

 prevailing westcrlv winds, and the littoral drift, separate 

 the finer particles from the sand and carry them eastwards, 

 and a microscopjc examination of samples obtained from 

 distances about a mile apart on this shore confirms this. 



This musical sand is of the Studland Bay type, and 

 near the Haven gives even better results than any I have 

 found tTiere. The occurrence of musical sands along this 

 particular shore through the conserving influence of the 

 groynes is an interesting fact, for their existence there 

 previously was very unusual, being only once noted in very 

 small quantitv during the last twentv years. 



Parkstone-on-Sea, July 4. Cecil Carus-Wilson. 



The Commutative Law of Addition, and Infinity. 



Referring to the review of Ililbert's " Grundlagen der 

 Gcometrie," on p. 394 of No. 2066 of M.\ture (June 3), 

 mav I point out that the commutative law of addition can 

 be proved without the help of any axioms at all, other than 

 those of general logic? The method, indeed, used by Peano 

 in 1889 (" Arithmetices Principia . . .," Turin, 1S89, p. 4), 

 which is only based on axioms of a general nature (such 

 as the principle of mathematical induction), and not on 

 such special laws as the distributive ones, appears in so far 

 superior to Hilbert's ; and, since all Peano's axioms 

 were proved in Mr. Russell's " Principles of Mathematics " 

 of 1903, Hubert's proof seems quite superseded. Further, 

 the difficulties arising out of Dedekind's proof of the exist- 

 ence of infinite systems can be avoided without the intro- 

 duction of " metaphysical " arguments about time and 

 consciousness (see Russell, Hibbcrt Journal, July, 1904, 

 pp. S09-12), as, indeed, vour reviewer seems to think 

 possible. But the connection of the fact that the existence 

 of an infinity of thoughts (which must be in time) with 

 Hamilton's idea that algebra was interpretable especially 

 in the time-manifold, just as geometry is in the space- 

 manifold, is not obvious. Philip E. B. Jouhdain. 



The Manor House, Broadwindsor, Beaminster, Dorset, 

 July 2. 



Neither Dr. Hilbert nor the reviewer make any sug- 

 gestion that the commutative law of addition is best proved 

 as a deduction from the laws of multiplication. But the 

 laws of multiplication are so often treated as deductions 

 from those of addition that it is interesting to have a case 

 of the converse procedure. The fact that both these opera- 

 tions and their laws have been treated independently and 

 in a strictly logical manner by Drdekind, Peano, and others 

 Is, of course, perfectly well known to all who have paid any 

 attention to this part of mathematics. Whether Dedekind's 

 critics have really avoided metaphysical arguments without 

 at the same time making metaphysical assumptions is a 

 question on which a difference of opinion is permissible. 



G. B. M. 

 NO. 2072, VOL. 81] 



THE THEORY OF CROOKES'S RADIOMETER. 



I HAVE noticed that the theory of this instrument 

 is usually shirked in elementary books, even the 

 best of them confining themselves to an account, and 

 not attempting an explanation.' Indeed, if it were 

 necessary to follow Ma.wvell's and O. Reynolds's cal- 

 culations, such restraint could easily be understood. 

 In their mathematical work the authors named start 

 from the case of ordinary gas in complete temperature 

 equilibrium, and endeavour to determine the first 

 efl'ects of a small departure from that condition. So 

 far as regards the internal condition of the gas, their 

 efforts may be considered to be, in the main, success- 

 ful, although (1 believe) discrepancies are still out- 

 standing. \\'hen they come to include the influence 

 of solid bodies which communicate heat to the gas 

 and the reaction of the gas upon the solids, the diffi- 

 culties thicken. A critical examination of these 

 memoirs, and a re-discussion of the whole question, 

 would be a useful piece of work, and one that may be 

 commended to our younger mathematical physicists. 



.Another way of approaching the problem is to select 

 the case at the opposite extreme, regarding the gas as 

 so attenuated as to lie entirely outside the field of the 

 ordinary gaseous laws. Some suggestions tending in 

 this direction are to be found in O. Reynolds's memoir, 

 but the idea does not appear to have been consistently 

 followed out. It is true that in making this supposi- 

 tion we may be transcending the conditions of e.x- 

 periment, but tlie object is to propose the problem in 

 its simplest form, and thus to obtain an easy and 

 unambiguous Solution — such as may suffice for the 

 purposes of elementary exposition, although the 

 physicist will naturall}' wish to go further. We 

 suppose, then, that the gas is so rare that the mutual 

 encounters of the molecules in their passage from the 

 vanes to the envelope, or from one part of the envelope 

 to another part, may be neglected, and, further, that 

 the vanes are so small that a molecule, after impact 

 with a vane, will strike the envelope a large number 

 of times before hitting the vane again. 



Under ordinary conditions, if the vanes and the 

 envelope be all at one temperature, the included gas 

 will tend to assume the same temperature, and when 

 equilibrium is attained the forces of bombardment on 

 the front and back faces of a vane balance one 

 another. If, as we suppose, the gas is very rare, the 

 idea of temperature does not fully apply, but at any 

 rate the gas tends to a definite condition which in- 

 cludes the balance of the forces of bombardment. If 

 the temperature be raised throughout, the velocities 

 of the molecules are increased, but the balance, of 

 course, persists. The question we have to consider 

 is what happens when one vane only, or, rather, one 

 face of one vane, acquires a raised temperature. 



The molecules arriving at the heated face have, at 

 any rate in the first instance, the frequencies and the 

 velocities appropriate to the original temperature. As 

 the result of the collision, the velocities are increased. 

 We cannot sav that thev are increased to the values 

 appropriate to the raised temperature of the surface 

 from which thev rebound. To effect this fully would 

 probably require numerous collisions. .'\ny general 

 increase in the velocity of rebound is sufficient to 

 cause an unbalanced force tending to drive the 

 heated surface back, as O. Reynolds first indicated. 

 If we follow the course of the molecules after collision 

 with the heated surface, we see that, in accordance 

 with our suppositions, they will return by repeated 

 collisions with the envelope to the original lower scale 

 of velocities before there is any question of another 

 collision with the heated face. On the whole, then, 



1 See for example Poynting and Thomson's " Heat," p. 150. 



