January i, 1903] 



NA TURE 



203 



inside the sphere, there is only one position for Q. But if P is 

 inside the cone, there are two. The value of V at P is the same 

 for both, given by (3) reckoned positive always. So the real 

 V at P is double as much. 



. If the speed varies, the values of 11 will usually be not the 

 same in the two positions, so the two partial V's must be 

 separately reckoned. But the speed and path may vary in such 

 a way that there are more than two positions of Q which are 

 the centres of waves which all arrive at P at the same moment. 

 When there are any number of electrons moving about in 

 given paths, the following will give a broad idea of the nature of 

 the problem. To find V at a fixed point P at the moment t. 

 Let at that moment a spherical surface expand from P at speed 

 v, not forward in time, but backward. In expanding from 

 radius o to 00 , it will cross the electrons one after another. 

 Take note of the times of passage, t } , / 2 , &c. (less than /), of 

 the charges and their velocities. Then 



V = 2 2 i d ) 



4?rR;i - (ujv) cos e\ 



where R„=z>(/- A,), and 8„ is the angle at Q„ between R„ and u„. 

 Similarly as regards the vector potential. 



When it is allowed to exceed v, the effect is to increase the 

 number of crossings of electrons. An electron crossed twice 

 counts as two electrons. 



The value of/,, is |i - [ujv) cos 9„} _1 . The vector u,, is the real 

 velocity of Q» at the moment /,,. Its apparent velocity, as 

 viewed from P at the moment /, is u„/„ or - R„. It has no 

 necessary resemblance to the real velocity, and may be positive 

 or negative. The dot here signifies differentiation to I at P. 



Talking of potentials, I am tempted to add a few words about 

 their King, the Pan-potential. In equation (1) above, let q 

 be not djd(vl), but any sort of complex time differentiator, for 

 example, if p = djdl, 



,f = (k + cp){g + np), 



which is the special form for electromagnetic waves in a con" 

 ductor. Then (2) is still the solution for a point source, and in 

 general 



V = 2 ^=pan/ ( 5) 



is the pan-potential due to the distributed source/; It is not 

 the complete solution, because e'' r has not been counted ; but 

 that is not wanted when there is no birrier to reflect. 



For instance, if C is impressed electric current, in a con- 

 ductor, the characteristic ot H, magnetic force, is 



(V 2 -<r)H= - curl C. (6) 



It follows by the above that 



H = pan curl C, (7) 



that is, the magnetic force is the pan-potential of the curl of the 

 impressed current. The operations pan and curl are inter- 

 changeable, so 



H = curl pan C, (8) 



H= curl A, 



if 



A= pan C. 



(Similarly v pan = pan v, and div pan = pan div.) 

 I worked out this problem for a fixed point source of im- 

 pressed current some time ago (" Elec. Pa.," vol. ii. p. 432) 

 without reference to the pan-potential. The operational solu- 

 tion there given, equation (258), represents either (7) or (8). The 

 algebrisation was also done. There is no advantage in using the 

 A function in this particular case ; it is, in fact, more difficult to 

 find A first and then derive H than to obtain H without A. Simi- 

 larly as regards E, the electric force. The second circuital 

 law derives it from the H equation, so that it is not required 

 to introduce * to supplement A. 



If the point-source is in motion, the pan-potential requires 

 Dopplerisation as well as the ordinary potential. But this does 

 not require explicit representation for continuously distributed 

 sources. For example, the electromagnetic circuital equations 



curl (H-h) = u div & + {k + cp)V, (9) 



curl (e-E) = w div fiH + (f+^)H, (10) 



where u, w, e, h are functions of position and time, have the 

 solutions 



E-e = panX, H-h = panY. (11) 



To prove this, and determine the nature of X and Y, it 

 suffices to put the characteristics of E.- e and H -h in the form 



NO. 1731, VOL. 67] 



( 1 ), if- having the more general later meaning. Now (9) and 

 (10) lead to 



(?--V 2 )(E -e) = - Vp - curl G -(g+ np)C, (12) 



(</--V-)(H-h)= -v<r+ curl C -(/!• + ,p)G, (13) 



where 



p = div (E-e), <r=div(H-h), (14) 



C = u div t-E + [k + cp)e, (15) 



G = w div i>M+(g+ixp)b.. (16) 



So X and Y are the right members of (12) and (13) as defined. 

 C is the impressed electric current, G the impressed magnetic 

 current. It will be seen that no separate determination of scalar 

 potentials is required, because they are already included in X 

 and Y. Oliver Heayiside. 



Recent Dust Storms in Australia. 



On November 11, 12 and 13, 1902, New South Wales and 

 Victoria experienced severe dust storms, apparently caused 

 by a mild cyclone travelling from the west, as the dust reached 

 here yesterday morning, the wind at the time being very light. 

 The atmosphere was so loaded with fine dust that the sun looked 

 dim and objects less than a mile away were quite indistinct, and 

 all furniture, even with doors and windows closed, became 

 coated with a fine grey deposit. 



Reports from vessels coming along the coast say that the sea 

 had a peculiar leaden colour ; and a remarkable appearance 

 was seen in Sydney Harbour yesterday morning. Crossing 

 the harbour from the north to the south side, immedi- 

 ately on getting in sight of the sun the wavelets between the 

 steamer and the sun showed streaks of brilliant light metallic 

 blue colour. This was intensified when the boat entered the 

 still glassy water of Sydney Cove, when the back of each ripple 

 caused by the steamer on the sunny side showed a sheet of the 

 same colour and that most brilliantly. The water where un- 

 disturbed was covered by a slight scum, which might either be 

 settled dust or a layer of mineral oil, but appeared more like 

 the former. The colour had not the iridescent appearance 

 caused by oil, as it was a uniform pile blue and only showed on 

 the back of the wavelets. 



It seemed to me that this was an exaggerated example of the 

 blue colour of water caused by finely-divided mineral matter 

 seen in glacier waters and those of the hot lakes of New Zea- 

 land, where the water has silica in suspension. 



Will. A. Dixox. 



97 Pitt Street, Sydney, November 14, 1902. 



About half-past four o'clock on the afternoon of November 12, 

 I noticed that the sky to the north and north-east, from the 

 horizon half-way to the zenith, had assumed an extraordinary 

 chocolate-brown tint, due to clouds of that colour which were 

 moving towards us from the north-west. Under these clouds, 

 and moving from the north-east, were ashy-grey patches of 

 stratus, streaked with fantastic dark lines resembling bows and 

 boomerangs. A few drops of rain which fell about five o'clock 

 were charged with brown, earthy matter, and at six o'clock a. 

 paper which was held in the rain became spotted all over with 

 brown blotches. 



This fact, and the colour of the clouds, led me to the con- 

 clusion that a tornado had taken place in the interior of 

 Australia, whirling the fine dust high into the upper regions of 

 the atmosphere, in which position it was carried over the Straits 

 and then descended with the rain. 



At 6.20 p.m. the solid matter was still descending, but in less 

 quantity ; at 6.30 there was a marked diminution ; and by ten 

 minutes to seven the rain was all but free from it. 



While the six o'clock shower was descending, one heard the 

 remark on all sides that "it was raining mud" ; those who 

 were unfortunate enough to have their week's washing hanging 

 out at the time were doomed to a second day at the wash-tub. 



This remarkable occurrence recalls the events of Black 

 Thursday, 1851, when Victoria was swept by tremendous bush- 

 fires ; leaves and portions of charred ferns were carried up to 

 great heights by the currents of heated air, wafted across Bass' 

 Straits and deposited upon our shores ; the sky was so darkened 

 by huge volumes of smoke that, although in the height of 

 summer, lamps had to be lit early in the afternoon. 



West Devonport, Tasmania, H. Stuart. Dove. 



November 14, 1902. 



