74 



NATURE 



[January 19, 1922 



Letters to the Editor. 



[The Editor does not hold himself responsible for opinions 

 expressed 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.] 



Generalised Lines of Force. 



Let me direct attention to a notable paper by Prof. 

 E. T. Whittaker, in the Proceedings of the Royal 

 Society of Edinburgh for November last. 



It has often been discussed whether electric or 

 magnetic lines of force were the more fundamental, 

 and which might be regarded as the "cause " of the 

 other; a discussion rather like the old controversy 

 as to the direction of vibration in a polarised beam 

 of light. Maxwell's theory satisfied the disputants 

 by making both directions equallj' important. 



In this paper Whittaker extends Faraday's theory 

 of lines of force to electromagnetic activities in 

 general, expressing the complex facts by aid of the 

 space-time continuum of Minkowski, whereby any 

 form of kinematics can be expressed as a sort of four- 

 dimensional statics. The solenoids of Faraday are 

 shown to be special cases of a more general kind of 

 surfaces, called calainoids, which reduce to ordinary 

 electric lines of force when the field is purely electro- 

 static, while they reduce to ordinary magnetic lines 

 of force when the field is purely magnetic ; so that the 

 Faraday electric and magnetic lines are not two dis- 

 tinct and (as it were) rival things, but two limiting 

 cases of the same thing. The essential solenoidal con- 

 dition — strength inversely as cross-section — is re- 

 tained if, instead of electric or magnetic force 

 separately, the contra-variant V(E"-H-) is employed, 

 and if the cross-section is tHat of a calamoid. 



Ordinary equipotential surfaces, whether electric or 

 magnetic, are seen to be special cases of an " electro- 

 potential " or "magneto-potential" [? ether-poten- 

 tial] surface, which reduces to one or other of them 

 whenever the field becomes static. These electro- 

 potential surfaces, in general, exist in the four-dimen- 

 sional world of space-time; but when the field is 

 static each surface is wholly contained within three- 

 dimensional space, and is an ordinary equipotential 

 surface. 



The property of the Faraday lines of force, that 

 they are everywhere perpendicular to the equi- 

 potential surfaces, is shown to be a case of the more 

 general theorem that the calamoids are everywhere 

 "half-orthogonal" to the electropotential surfaces; 

 (half-orthogonality being the four-dimensional ana- 

 logue of three-dimensional perpendicularity) 



In electrostatics, the total strength of all the Fara- 

 day tubes which issue from a closed surface contain- 

 ing no electric charge is zero; similarly, in general 

 radiation-fields, the total strength of all the calamoids 

 which cross a closed surface is zero. This theorem 

 provides an intuitive geometrical integration of the 

 Maxwell-Lorentz equations of the electromagnetic field. 



There are also elaborated generalised "divergence " 

 and "curl " theorems, with a certain kind of abso- 

 luteness about them, since they are independent of 

 the motion of any observer. 



For much fuller and more trustworthv information 

 Prof. Whittaker's paper must be referred to. He 

 said something about it in Section A at the recent 

 meeting of the British Association in Edinburgh, but 

 I, for one, did not understand his meaning then, in 

 the rush of Sectional procedure. The object of the 

 present summary is merelv to direct earlv attention 

 to a paper which cannot be long overlooked. 



December 28. Oliver Lodge. 



NO. 2725, VOL. 109] 



Units in Aeronautics. 



Write the usual formula in aeronautics, 



s Viooy 



in the Hospitaller notation, and the practical airman 

 recognises that the resistance R, lb, over a surface S, 

 ft^, is at the rate 23-7 lb/ft* at a normal velocity 

 of 100 f/s ; and so on for any other velocity V, on the 

 law of the square. 



The airman pays no heed to any units except his 

 foot and pound, and he has no use for any of the 

 elaborate explanations of Mr. A. R. Low. 



The factor 237 will be the result of experiment in 

 the air-channel, reduced for air of standard conditions 

 of barometer and thermometer on the ground. 



But the air density is never measured in any of 

 these experiments, and it is doubtful if the measure- 

 ment has ever been carried out in any aeronautical 

 laboratory. 



In the early history of the Royal Society " weighing 

 the air " was a favourite research. Charles II. bet 

 Buckingham fifty guineas to one he would demonstrate 

 the compression of air in his hollow walking-cane ; 

 but the other story of the fish in a bucket of water 

 cannot be traced further back than Whateley, who is 

 supposed to have invented it. 



The air density arises in the formula of the treatise 

 on aerodynamics on the idea that the formula is the 

 expression of Newton's assumption that the resistance 

 is due to the impact of inelastic air particles, as if 

 air could be treated as a cloud of dust ; and then at 

 an air density w, lb/ft', or better for calculation in 

 thermo-aerodynamics, at a specific volume the 

 reciprocal C = i/w, ft'/lb, Newton's formula becomes 



.R=e.V^=r, IbTt- or ^ = 2^.7. A = V^l 



S g gC S 2g 



In this treatment the Equation of Continuity is 

 ignored ; the air particles should stop dead and fall 

 down in a heap at the foot of the aeroplane, to be 

 swept up as dust. 



Thus the mysterious factor 000237 of the treatise 

 on aerodynamics is the equivalent of 7v/g, and with 

 ^ = 322, f/s", this makes ^ = 00763, lb/ft', = 13-1, 

 ft'/ib, so that this standard air bulks 13 cubic feet 

 to the pound, in round numbers. 



Another wav of expressing the law is to write it 

 in the equivalent form 



so that H is the velocity at which normal resistance 

 is I lb per square foot ; then on the figures above 

 H = 20-5 f/s, and this may be replaced by 20 in 

 round numbers for practical calculation, making 



.R = 25fVY. 

 S H 100/ 



Flying over the ocean the velocity would be ex- 

 pressed in knots, K, and with 12 knots the equivalent 

 of 20 f/s the formula is ■ '"" 



hC^oM^X -"" 



sim])le numbers easily remembered. 



In all these calculations Perry's dictum must be 

 respected : that the accuracy of a formula is only 

 the accuracy of its most inaccurate part. 



Here the index 2 of the velocity, adopted for 

 simplicity of calculation, is the part most subject to 

 doubt, and then at this rate of the quadratic law 

 the above numbers, 237, 24, and 25, are all equally 



