May 13, 1909] 



NATURE 



309 



the time of experiment in the uranium-radium series. It 

 is easily shown, provided all the periods are long, that 

 the initial rate of production must be proportional to the 

 same power of the time as the number of substances 

 (including uranium itselfj in the series before radium. 

 These experiments, therefore, indicate that there is only 

 one intermediate substance in the uranium-radium series 

 with a long period of life. Assuming what is probable, 

 but not yet known, that the present law and rate of 

 production will be continued in the future, it is possible 

 to lix the period of average life of the intermediate sub- 

 .siance from the existing data, with a margin of un- 

 certainty probably not greater than 20 per cent. Ruther- 

 ford has shown that the initial production of radium in 

 grams from a kilogram of uranium is equal to 

 6xio-''A.T-, where i/\ is the period of average life of the 

 intermediate body in years, and T is the time in years. 

 This gives for the average life of the intermediate body 

 the period of 10,000 years. This is about four times that 

 of radium itself, and there should exist in uranium 

 minerals about 1-36 milligrams of the substance per kilo- 

 gram of uranium. The initial rate of production over the 

 first two years appears less than that calculated, as though 

 another intermediate substance of period of the order of 

 two years existed in the series ; but greater refinements 

 will be necessary before this can be definitely proved by 

 experiments of this character. 



Frederick Soddy. 

 Physical Chemistry Laboratory, University of 

 Glasgow. 



Wave Motion and Bessel's Functions. 



The property, enunciated by Dr. Johnstone Stoney, 

 according to which any wave motion can be regarded as 

 built up of a combination of plane waves, may be used 

 with advantage for a verification of the formulse for the 

 solutions of Bessel's equation in the form of definite 

 integrals. 



Consider, for example, the hydrodynamical problem of 

 circular waves about the axis of e in a liquid of uniform 

 depth extending from s = o up to the free surface 2 = /j. 

 Imagine the wave motion to be built up by the super- 

 po^iition of a continuously infinite number of plane waves, 

 symmetrically distributed about the axis of s. By taking 

 a to denote the angle which the normal to any wave 

 front makes with the radius vector to any point, and by 

 integrating the expression for the velocity potential of the 

 corresponding train of plane waves with respect to o, we 

 get the expression 



=-v:. 



cos m\r cos a - 1'^ + c] cosh iiiz da, 



z"=^^ tanh mh. 



The above expression for ^ being a solution of Laplace's 

 equation, it follows that 



/: 



cos w (>■ COS a) da and / sin m (r cos o) da 



/:.' 



are solutions of the corresponding Bessel's equation in r. 



Next, taking an unsyminetrical distribution of plane 

 waves, and confining attention to the particular case in 

 which the phase relative to the origin is independent of 

 the direction, the amplitude between the directions a and 

 a + da being F(a)(io, we find for the potential at the point 

 (r. ft, z) the expression 



. = /F{a 



) cos m \r cos (a - 9) - 1// + e! cosh luz da. 



the integral being taken between limits for a differing by 

 27r. By writing o— e=a?, and suitably choosing the limits, 

 we find 



<>.= [" F(9H 



cos 1)1 \v COS oi-vt-'ri) cosh }iiz da. 



and taking the particular cases of F(o) = cos or sin »a, we 

 obtain the solutions of Laplace's equation 



cosh mz cos or sin iiB j cos or sin nu cos or sin //; (/• cos (e)da 



NO. 2063, VOL. 80] 



leading to solutions of Bessel's equation of order n, 

 namely, 



/ cos or sin iiai cos or sin m {r cos a) da. 



G. H. Bryan. 



University College of North Wales, Bangor. 



Dew-Fonds. 



Like " E. A. M." in N.\ture of April 22, I have always 

 been extremely sceptical about Mr. Hubbard's theory of 

 dew-ponds since it first appeared in " Neolithic Dew-ponds 

 and Cattle-ways." My own experience of lakes and ponds 

 is thatj.they lose their heat slovi'ly, and that, after radia- 

 tion it- has : set in at night, they indicate a much higher 

 temperature than the ground adjoining or the air above. 



It has been a matter of frequent observation on Coniston 

 Lake in summer that, after a night of heavy dew, the 

 bottoms of the boats inside were perfectly dry, whilst 

 the gunwale was covered , with moisture, showing that 

 the portion of the boat in contact with the water had been 

 raised to a temperature above the dew-point. Prof. Miall 

 and myself a few years ago made a special expedition 

 to the Berkshire downs, in the ■ neighbourhood of Wan- 

 tage, to determine the temperature of the dew-ponds, and 

 we found precisely the same thing, that is to say, the 

 water at night was warmer than the air. It is impossible, 

 therefore, that dew could deposit on the ponds under these 

 conditions. 



Moreover, as " E. A. M." points out, it is inconceivable 

 that the clay or straw in a full dew-pond can have much 

 connection with the temperature of the water. My own 

 conviction is that the straw is merely used to bind the 

 clay, and the bed of clay above the chalk serves no other 

 purpose than to make the pond bottom water-tight. No 

 satisfactory explanation of dew-ponds has yet been pro- 

 pounded, and, as your correspondent says, " there is room 

 for more experiment." I have seen no reference to what 

 may, I think, constitute one important factor in the re- 

 plenishment of dew-ponds. Aitken has shown that the 

 greater portion of the moisture deposited as dew is derived 

 from the ground and not from the air, and in this con- 

 nection it should be remembered that the chalk, on which 

 the ponds are usually constructed, absorbs water like a 

 sponge. Consequently, any lowering of temperature may 

 cause a heavier dew or mist formation than on less 

 absorbent material. Seeing that many of the ponds lie 

 quite exposed on the very summit of the downs, drainage 

 of dew cannot feed them, and it seems probable that mist 

 may in some cases play a more important r6le than dew. 



J. B. Cohen. 



The Imperial Side of the Fuel Ouestion. 



The article in Nature of May 6, and Sir W. Ramsay's 

 comment upon it, direct attention to a most important 

 economic question. It has often crossed my mind that by 

 a simple legislative enactment a marked saving might be 

 effected in our factory consumption of coal. If Parlia- 

 ment would enact that after a given year, say 1920, a con- 

 siderable penalty should be payable by the owner of any 

 factory where the consumption of fuel coal exceeded ij lb. 

 per hour per indicated horse-power, it is probable that 

 almost all factories would by that date be improved up 

 to that level of efficiency. 



It is probable that the average efficiency of steam 

 plants is only about 3 lb. per indicated horse-power hour, 

 and your article shows the factory consumption to be 

 about si.xty-one million tons per annum. On those figures, 

 halving the consumption on the above lines would save 

 about thirty million tons a year. The modernisation of 

 plant involved would pay for itself (from the factory 

 owner's point of view) in a very few years, and so would 

 be a remunerative investment, so much so that financing 

 the change should be within reach of even the weaker 

 firms. 



The thirty million tons I suggest might be saved i! 

 more than 11 per cent, of the production (figures of 1907), 

 so that the saving is well worth the attention of all who 

 are concerned to conserve our coal, and I trust that the 

 idea may be pressed forward in influential quarters. 



Manchester, May 10. .iKrthvr McDougall. 



