452 



NA TURE 



{March 7, 1889 



A closer consideration of the figures shows that the direction 

 of the force changes from instant to instant for such points as 

 lie either in the axis of 2 or in the plane xy. If we represent 

 the force at a point, therefore, in the customary way by a line, 

 the end point of this line oscillates, not indeed in a straight line, 

 but in an ellipse. In order to see whether there are points for 

 which this ellipse approximates to a circle, in which, therefore, 

 the forces go through all the directions of a windrose without 

 important change of magnitude, let us superpose two of the 

 representations expressing times which differ by ^T ; for instance, 

 Figs. I and 3, or 2 and 4. 



For the points we seek, the lines of the one set must plainly 

 •cut those of the other system orthogonally, and the distances of 

 the lines of the one figure must be equal to those of the other. 

 The small quadrangles formed by the superposition of the two 

 systems must therefore be squares for the sought points. 



There may be now remarked, in actual fact, a region of the 

 kind sought : it is represented in Figs, i and 2 by circular 

 arrows, whose directions give at once the direction of the rota- 

 tion of the force. The dotted lines are inserted for convenience ; 

 they belong to the line system of Figs. 3 and 4, 



One finds, moreover, that the force exhibits the behaviour 

 here described, not only at the specified points, but also in the 

 whole strip-formed region which, spreading out from those 

 points, forms the neighbourhood of the z axis. Nevertheless, 

 the magnitude of the force decreases so quickly in these direc- 

 tions, that only in the points above-mentioned can its singular 

 behaviour be important. 



The system of forces now described and required by theory 

 can be quite well recognized in an incomplete observation, 

 not hitherto indicated by theory, which I formerly described 

 {JVted. Ann. xxxiv. p. 155, 1888). One cannot, indeed, explain 

 everything about those experiments, but one can get the main 

 points correctly. 



By both experiment and theory the distribution of force in the 

 neighbourhood of the oscillator is chiefly an electrostatic dis- 

 tribution. By both experiment and theory the force spreads out 

 chiefly in the equatorial plane and decreases in that plane at first 

 quickly, afterwards slowly, without being zero at a mean distance. 

 By both theory and experiment the force, in the equatorial plane, 

 in the axis, and at great distances, is of constant direction and 

 varying magnitude, while at intermediate points it changes its 

 magnitude but little and its direction much. The correspondence 

 between theory and those experiments only breaks down in this, 

 that at great distances, according to theory, the force remains 

 always normal to the straight line through the source, while by 

 experiment it appears to be parallel to the oscillator. For the 

 neighbourhood of the equatorial plane where the forces are 

 strongest this follows from the equations too, but not for directions 

 which lie between the equatorial plane and the axis. I believe 

 that the error is on the side of experiment. In these experiments 

 the direction of the oscillator was parallel to both the main walls 

 •of the laboratory, and the component of the force which was 

 parallel to the oscillator might be thereby strengthened in 

 proportion to the normal components. 



I have therefore repeated the experiment with a different 

 arrangement of the primary oscillator, and found that with 

 -certain arrangements the result corresponds with theory. I did 

 not attain an exact result, but found that at great distances, and 

 in regions of small intensity of force, disturbances due to the 

 boundary of the space available were already too considerable to 

 permit a safe verdict. 



While the oscillator is at work, the energy vibrates in and out 

 ■oi the spherical surfaces surrounding the origin. More energy 

 goes out, however, through any spherical surface during an 

 -oscillation than comes back ; and indeed the same excess 

 •quantity goes through all spherical surfaces. This extra quantity 

 represents the loss of energy during the period of swing due to radia- 

 tion. We can easily calculate its value for a spherical surface 

 whose radius, p, is so great that it is permissible to employ a 

 simplified formula. Thus the energy going out of the spherical 

 ^one between d and + dB \n the time dt will be — 



p 



2iTp sin e pde dt. (Z sin 



4irA 



R cos e). 



from if = o to T, we get, as the energy going out through tht 

 whole sphere during every half complete swing,— 



l^E^PnfinT 



Let us try to obtain an approximate estimate of the amount 

 of this corresponding to our actual experiments. In those 

 we charged two spheres of 15 centimetres radius in opposite 

 senses up to a spark length of i centimetre about. We may 

 estimate the difference of potential between these spheres as 

 120 C.G. S. electrostatic units, so each sphere was charged to 

 half this potential, and its charge was therefore E — 900 

 C.G.S. units. 



The total store of energy which the oscillator originally possessed 

 amounted to 60 ;< 900 = 54,000 ergs, or 55 centimetre-grammes. 

 The length of the oscillators, moreover, was i metre approxi- 

 mately, and the wave-length was about 480 centimetres. 



So the loss of energy in half a swing comes out about 2400 ergs. 

 It seems, therefore, that after eleven half-swings one-half of the 

 energy must have gone in radiation. The quick damping which 

 the experiments made manifest was therefore necessitated by 

 radiation, and could not be prevented even if the resistance of 

 conductor and spark were negligible. 



A loss of energy of 2400 ergs in i "5/100,000,000 of a second 

 means a performance of work equal to 22 horse-power. The 

 primary oscillator must be supplied with energy at at least this 

 rate if the oscillation is to be permanently maintained at constant 

 intensity in spite of the radiation. During the first few oscilla- 

 tions the intensity of the radiation at about 12 metres distant 

 from the vibrator corresponds with the intensity of solar radia- 

 tion at the surface of the earth. 



( To be continued. ) 



GENERAL EQUATIONS OF FLUID MOTION. 



'T*HE general equations of the motion of a fluid can all be com- 

 -*■ prehended in a single form, which seems to be deserving 

 of special notice. 



Taking the ordinary notation, u, v, w, for the velocity-com- 

 ponents at any point, P, of the fluid at any instant, and denot- 

 ing the components of vortical spin at the point by wj, 01^, cog, 

 the usual Cartesian equations can be at once put into the form — 



t + ^(^?' + \j) + 2(.t.<.,-z,a,3) = X, 



and two analogues, tj being the resultant velocity. If through 

 the point P we draw any curve whatever, the direction-cosines 

 of whose tangent are /, m, n, and multiply the above and its ■ 

 two analogues, respectively, by /, m, n, we obtain by addition 

 the equation- 



(a) 



dt ds 



Putting into this the values of Z, P, and R, which are 

 j)roper for great distances, and integrating from = o to tt, and 



in which s stands for the component of velocity along the tangent 

 to the curve, U = ^^^ -I- | — , S = component of external force- 

 intensity along the tangent, and A is the volume of the tetra- 

 hedron formed by the vector drawn at P to represent q, the 

 resultant velocity, the vector drawn to represent n, the result- 

 ant vortical spin, and the vector representing a unit length along 

 the tangent to the curve at P. (Strictly speaking, the notation .s- 

 is not a good one, but it is the best that presents itself.) 



This equation (a) is that which I propose, as typical of all 

 fluid motion, and as including all the special Cartesian equations 

 in current use. 



Some simple- results follow at once for the case of steady 

 motion. Thus, if we integrate (a) between any two points, A, 

 B, of the curve, 



Ub -- Ua + 12^ Ads = fsds (i) 



where Ub and Ua are the values of U at B and A. 



Now, in particular, if the curve drawn at P is a stream-line, 

 A = o at every point of it ; also, if the curve is a vortex-line, 

 A = o at every point, and we have the simple result. 



Ub- Ua 



= fsds 



(2) 



