202 



NA TURE 



[January i, 1903 



the assay of the ores of gold, silver, copper, lead, tin, 

 zinc, iron, manganese and chromium, and the estimation 

 of the impurities usually met with in these metals. There 

 are also sections devoted to the analysis of fluxes, re- 

 fractory materials, slags, fuels, &c, and of white alloys, 

 iron alloys and copper alloys. This enumeration is 

 enough to show that an enormous amount of ground is 

 covered, and in such a small book there is, of course, no 

 room to explain the principles on which the directions 

 are based. The directions themselves, however, are 

 clear and generally accurate. Among the mistakes and 

 omissions which have been noticed, the neglect to re- 

 oxidise the lead reduced by the filter-paper in the ignition 

 of lead sulphate and an erroneous method of calculating 

 the gold contained in ores may be mentioned. The 

 modern method of adding zinc acetate instead of soda 

 and acetic acid in the iodide copper assay is not men- 

 tioned, and electrolytic methods generally are ignored. 

 The book should be useful to students in polytechnics 

 and other evening schools. 



First Stage Mathematics. Edited by W. Briggs. Pp. 



vii + 186. (London: Give, 1902.) 

 This volume of "The Organised Science Series" deals 

 with the geometry and algebra required for Stage i. 

 Mathematics in the South Kensington examinations. 

 In section i., the text of Euclid i. is strictly followed, 

 with occasional notes and explanations. Useful sum- 

 maries of the propositions are given, also several sets of 

 easy exercises and a number of miscellaneous riders. A 

 few additional propositions are proved, and hints are 

 given on writing out proofs. 



In section ii. (algebra), in one or two places the 

 reasoning does not appear quite satisfactory. For 

 example, on the sign of a product, on p. 21 : — "Suppose 

 4-2 x -3 or -2X+3. Evidently the product will not 

 be the same in either of these cases as in 4-2 x +3. 

 Therefore we assume that +2X —3= -6 and -2X +3 

 = — 6. Therefore, when one term has a phis sign and 

 the other term has a minus sign the product is minus. 

 Again, suppose — 2X -3. This is different from the 

 last two cases, and we assume that — 2 x — 3=4-6. There- 

 fore, when two terms with minus signs are multiplied 

 together the product is plus. From these results we can 

 inler the rule of signs/' A statement of this kind almost 

 inevitably tends to fog the mind of a student. 



The use of the word sum (as on p. 52) in any other 

 than its exact algebraic meaning, in a text-book for 

 beginners, is objectionable. 



There are numerous easy exercises in algebra, also 

 arithmetical questions from previous papers. 



The book is well printed, and the figures in the 

 geometry are clearly drawn. 



Preparatory Lessons in Chemistry. By Henry W. Hill. 

 Pp. v+122. (London : Allman and Son, Ltd.) Price is, 



The order of treatment in this little book represents the 

 method of teaching chemistry more common twenty 

 years ago than now. Before being set to examine for 

 himself easy familiar chemical changes in a scientific 

 manner, the beginner is expected by the author to be 

 able to understand such subjects as atoms and mole- 

 cules, formula; and equations, and similar matters much 

 more suitable for students at a later stage of work. 

 Several better books for beginners in chemistry are avail- 

 able. 



My Dog Frizzie and Others. By Lady Alicia Black- 

 wood. Pp.44. (London : Operative Jewish Converts' 

 Institution). Price 4^. 



These are simple, interesting stories concerning the 

 habits and character of a pet dog. The tales may en- 

 courage children to study animals intelligently. 



NO. I 73 I, VOL. 67] 



LETTERS TO THE EDITOR. 



[ The Editor does not hold himself responsible for opinions ex- 

 pressed 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.] 



Sound Waves and Electromagnetics. 

 The Pan-potential. 



The photographs taken some years ago by Prof. Boys of 

 flying bullets showed the existence of a mass of air pushed 

 along in front of the bullet. Is there anything analogous to 

 this in the electromagnetics of an electron ? Suppose, for 

 example, that an electron is jerked away from an atom so 

 strongly that its speed exceeds that of light. Then it will slow 

 down by reason of the resisting force to which I have shown it 

 is subjected. So long as its speed is greater than that of light, 

 it is accompanied in its motion by a conical wave. The question 

 is whether there is any disturbance ahead of the electron, close 

 to it, as in the case of a bullet moving through the air. It is a 

 question of fact, not of theory. When Maxwell's theory shows 

 that there is no disturbance in front of the electron, that is only 

 because it is virtually assumed to be so at the beginning, by the, 

 assumption that the ether continues fixed when the electron 

 traverses it. 



Apart from this detail, the analogy between the conical sound 

 wave and the conical electromagnetic wave is interesting in 

 connection with C. A. Bjerknes's theory of pulsations in a 

 liquid, as developed by V. Bjerknes in his " Vorlesungen uber 

 hydrodynamische Fernkrafte nach C. A. Bjerknes' Theorie. " 

 The liquid is incompressible, and is set into a pulsating state 

 by pulsating sources, and the result shows remarkable analo- 

 gies with electric and magnetic phenomena when they are 

 static. 



Now if the liquid is compressible, the results must be ap- 

 proximately the same provided the pulsations are not too 

 quick. But if very rapid, and the compressibility be sufficient 

 to lower the speed of propagation sufficiently, new phenomena 

 will become visible with pulsating sources, like sound waves, 

 and the question is how far they are analogous to electro- 

 magnetic phenomena ? 



Here, for example, is an interesting case. Let / be the 

 density of the source, such that (if f=d/d K vt)), 



( V 2_ ? S)V=_/ (I) 



is the characteristic of the velocity potential V, so defined that 

 - v"Y is the velocity. Then f signifies the amount of fluid 

 (unit density) generated per unit volume per second and 

 diverging outward. Then, for a point source of strength Q, 

 the V it produces is 



\' = ( —0^''' (2) 



4irr 47rr 



at distance ;•. This is equivalent to Rayleigh's account of 

 Helmholtz's spherical waves from a centre ("Theory of 

 Sound," vol. ii.), except in the interpretation of f or Q, which 

 I do not altogether understand in that work. 



Q is a fluctuating function of the time in the above in the 

 acoustic application, though, of course, fluctuation is not 

 necessary in the ideal theory. Now if the source Q moves 

 through the air with velocity u, the potential becomes 



V= 9 , (3) 



4irr\i - (u/v) cos9[ 



if is the angle at Q between r and u at the proper moment. 



This equation therefore expresses the theory of a very small 

 pulsating source moved through the air, and is so far very like 

 that of an electric charge Q (which does not pulsate) moved 

 through the ether. The analogy does not continue in details, 

 when, for example, we compare velocity with electric displace- 

 ment. The electromagnetic theory is more involved. 



When u exceeds v, equation (3) is no longer the complete 

 solution. If u is less than v, there is just one and only one 

 position of Q at a given moment where it is, so to speak, in 

 communication with P, the point where V is reckoned. But 

 when u>v, there may be just one point, or two, or there may 

 be any number. Thus, if the source Q starts at moment 

 t = o from a certain point, and then moves steadily in a straight 

 line, the wave front is conical, with a spherical cap, or 

 spherical, with a conical spike, Q being at the apex. If P is 



