54 



NA TURE 



[May 2 1, 1903 



Society, 1 should be glad if you could find space in Nature 

 for an account of them. 



The dust consisted essentially of ferruginous sand, chalk, 

 and silicates of alumina, alkalis, lime and magnesia, mixed 

 with a certain quantity of organic matter and with an 

 appreciable proportion of lead. 



The last-named substance is probably due to the sample 

 having been collected from a leaded roof. It may either 

 have been scraped off during the taking of the sample, or, 

 possibly, cut from the leads by the impact of sand particles 

 driven against the roof by a high wind. Traces of tin and 

 arsenic were also present in the sample ; these were probably 

 contained as impurities in the lead. 



The detailed results of the analysis are as follows : — 



(Substance dried at 100° C. before analysis.) 



Per cent. 



Loss on heating to redness 11-28 



Lead, calculated as oxide 3.31 



Arsenic 001 



Tin Traces 



After deducting the lead, tin and arsenic as being prob- 

 ably adventitious, the remainder of the sample is made up 

 of the following constituents : — 



Silica 

 Alumina ... 



Iron oxide 



Lime 

 Magnesia 



Alkalis / J°f '""} o'^'de.^ 

 (^ Potassium oxide 



Carbonic acid 



Water and organic matter 



Per cent. 



45-94 

 18-35 

 6-57 

 864 

 1-86 

 116 

 230 

 610 

 908 



The organic matter contained 2- 19 per cent, of carbon 

 and 016 per cent, of nitrogen, the two representing, prob- 

 ably, between 3 and 4 per cent, of organic constituents. 



After being heated to redness, 3330 per cent, of the sample 

 was found to be soluble in hydrochloric acid, the dissolved 

 portion including practically the whole of the lead, with 

 the traces of tin and arsenic. Again deducting those 

 elements, the dissolved constituents were as follows : — 



Silica 



Alumina ... 

 Iron oxide 

 Lime 

 Magnesia 

 Alkalis 

 Carbonic acid 



Per cent. 

 0-64 

 II-20 



5-43 



8-19 

 113 

 1.46 

 348 



31-53 



Thus about one-third of the sample is dissolved by hydro- 

 chloric acid, including the greater part of the alumina, iron, 

 lime and magnesia, but only a small fraction of the silica. 



Dilute acetic acid readily dissolved out the greater part 

 of the lime, with liberation of carbonic acid gas. Water 

 alone dissolved practically nothing from the sample except 

 minute traces of lime. These results show that most of 

 the lime is present in the sample in the form of chalk. 



One or two particles of metallic lead were detected in 

 the sample, together with others partly oxidised and 

 carbonated. 



It has been surmised by Dr. Mill and others that the 

 sand which accompanied the storm of February 22, and was 

 observed to fall in a great number of places in this country 

 as well as on the Continent, was originally derived from the 

 African deserts. 



It would be interesting in this connection to compare its 

 characters with that of the dust, also presumably of African 

 origin, which was observed to fall in the neighbourhood 

 of Taormina, by Sir Arthur Riicker, and was made the 

 subject of an interesting communication to Nature by Prof. 

 Judd about a year ago. T. E. Thorpe. 



Government Laboratories, London, W.C. 



NO. 175 1, VOL. 68] 



The Undistorted Cylindrical Wave. 

 The receipt of a paper by Prof. H. Lamb, " On Wave 

 Propagation in Two Dimensions " {Proc. Lond. Math. Soc, 

 vol. XXXV. p. 141), stimulates me to publish now a con- 

 densation of a portion of a work which will not be further 

 alluded to. I once believed that there could not be an 

 undistorted cylindrical wave from a straight axis as source. 

 But some years ago the late Prof. FitzGerald and I were 

 discussing in what way a plane electromagnetic wave 

 running along the upper side of a plane conducting plate, 

 and coming to a straight edge, managed to turn round to 

 the other side. Taking the wave as a very thin plane slab, 

 one part of the theory is elementary. The slab wave itself 

 goes right on unchanged. Now Prof. FitzGerald specu- 

 latively joined it on to the lower side of the plate by means 

 of a semi-cylindrical slab wave. I maintained that this 

 could not possibly work, because the cylindrical wave 

 generated at the edge was a complete one, causing back- 

 ward waves on both sides of the plate. Moreover, it was 

 not a simple wave, for the disturbance filled the whole 

 cylindrical space, instead of being condensed in a slab. It 

 was in the course of examining this question that I arrived 

 at something else, which I thought was quite a curiosity, 

 namely, the undistorted cylindrical wave. 



Maxwell's plane electromagnetic wave consists of per- 

 pendicularly crossed straight electric and magnetic forces, in 

 the ratio given by E = ;uz/H. Thinking of a thin slab only, it 

 travels through the ether perpendicularly to itself at speed 

 V, without any change in transit. I have shown that this 

 may be generalised thus. Put any distribution of electrifi- 

 cation in the slab, and arrange the displacement D in the 

 proper two-dimensional way, as if the medium were non- 

 permittive outside the slab. Then put in H orthogonally, 

 according to the above mentioned rule, and the result is the 

 generalised plane wave, provided the electrification moves 

 with the wave. Otherwise, it will break up. Another way is 

 to have the electrification upon fixed perfectly conducting 

 cylinders arranged with their axes parallel to the direction 

 of propagation. 



Now the first kind of plane wave has no spherical 

 analogue, obviously. But I have shown that the other kinds 

 may be generalised spherically. Put equal amounts of 

 positive and negative electrifications on a spherical surface 

 arranged anyhow. Distribute the displacement in the proper 

 way for a spherical sheet, as if constrained not to leave 

 it. Then put in H orthogonally as above. The result con- 

 stitutes an undistorted spherical electromagnetic wave, pro- 

 vided the electrification moves radially with the wave, and 

 attenuates in density as its distance from the centre in- 

 creases, in the proper way to suit E and H. This attenu- 

 ation does not count as distortion. Similarly, the other sort 

 of generalised plane wave may be imitated spherically by 

 having conical boundaries. 



But when we examine the cylinder, there is apparently 

 no possibility of having undistorted waves. For with a 

 simple axial source it is known that if it be impulsive, the 

 result is not a cylindrical impulse, but that the whole space 

 up to the wave front is filled with the disturbance. It is 

 easy to see the reason, for any point within the wave front 

 is receiving at any moment disturbances from two points 

 of the source on the axis, and there is no cancellation. And 

 if the source be on a cylindrical surface itself, producing an 

 inward and an outward wave, the whole space between the 

 two wave fronts is filled with the disturbance. 



How, then, is it possible to have an undistorted wave 

 from a straight line source? By not arguing about it, but 

 by showing that it can be done. The reason will then 

 come out by itself. As the solution can be easily tested, it 

 is only necessary to give the results here. Take plane 

 coordinates r and 0. Let the magnetic force be perpen- 

 dicular to the plane, of intensity H. Let Z be its time- 

 integral, then 



Z = ^/(z,/ - r\ H = ^-2i^/'(z,/-.), (I) 



expresses the magnetic field, / being an arbitrary function. 

 Now the displacement D is the curl of Z. So if E, is the 

 radial component of E, and E3 the tangential component, in 

 the direction of increasing 9, we have the electric field 

 given by 



■ yi.v sin \9 



/, 



Tj. _tiv cos \9 f, , fi-v cos \e r 



^2 :,r~-J +~^rr^^- 



(2) 



