April 2, 1896] 



NATURE 



509 



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. '\ 



Velocity of Propagation of Electrostatic Force. 



As we may have to wait some time for the experimental solu- 

 tion of Lord Kelvin's very instructive and suggestive problem 

 -concerning two pairs of spheres charged with electricity (see 

 Naiure of February 6, p. 316), it may be interesting to see 

 what the solution would be from the standpoint of existing 

 electrical theories. 



In applying Maxwell's theory to the problem, it will be con- 

 venient to suppose the dimensions of both pairs of spheres very 

 small in comparison with the unit of length, and the distance 

 between the two pairs very great in comparison with the same 

 unit. These conditions, which greatly simplify the equations 

 which represent the phenomena, will hardly be regarded as 

 affecting the essential nature of the question proposed. 



Let us first consider what would happen on the discharge of 

 (A, B), if the system {c, d) were absent. 



Let Wfl be the initial value of the moment of the charge of the 

 system (A, B), (this term being used in a sense analogous to 

 that in which we speak of the moment of a magnet), and m the 

 value of the moment at any instant. If we set 



m - W), (I) 



and suppose the discharge to commence when t = o, and to be 

 • completed when t — h, v/q shall have 



F(/) = Wq when t<o, .... (2) 



and 



Y{t) = o when i> h, .... (3) 



Let us set the origin of coordinates at the centre of the 

 system (A, B), and the axis of x in the direction of the centre of 

 the positively charged sphere. A unit vector in this direction 

 we shall call i, and the vector from the origin to the point 

 considered p. At any point outside of a sphere of unit radius 

 about the origin, the electrical displacement (3)) is given by the 

 vector equation 

 4»2) = [3r-^F(t - cr) + 3cr-*F'(t - cr) + c'^r-^¥"(t - cr)]xp 



-[r-3F(^ - cr) ■{■ cr-^F'(t - cr) + c^r-^F"(t - cr)]i, . (4) 

 where F denotes the function determined by equation (i), F' 

 and F" its derivatives, and c the ratio of the electrostatic and 

 electromagnetic units of electricity, or the reciprocal of the 

 velocity of light. For this satisfies the general ecjuation 



- v^I) = c^d^^/dt^ (5) 



as well as the so-called "equation of continuity," and also 

 satisfies the special conditions that when ^ < o 



4*2) = /«o(3^"'xp - ^~'^i) 

 outside of the unit sphere, and that at any time at the surface of 

 this sphere 



4»JD = w(3xp - i), 

 if we consider the terms containing the factor c as negligible, 

 when not compensated by large values of r. That equation (4) 

 satisfies the general conditions is easily verified, if we set 



« = r-i F(/ - cr), (6) 



and observe that 



- v^u = c'^cPuldt", (7) 



and that the three components of 2) are given by the equations 

 4^/ = - d^uldf - d-u/dz- . . . ) 



^ng = d^u/dxdy [(8) 



^irh = d^u/dxdz ) 



Equation (4) shows that the changes of the electrical displace- 

 ment are represented by three systems of spherical waves, of 

 forms determined by the rapidity of the discharge of the system 

 (A, B), which expand with the velocity of light with amplitudes 

 diminishing as r~*, r'-, and r'^, respectively. Outside of these 

 waves, the electrical displacement is unchanged, inside of them 

 it is zero. 



If we write (with Maxwell) -d^/dt for the force of electro- 

 dynamic induction at any point, and suppose its rectangular 

 components calculated from those of - d'')£)/dt- by the formula 



NO. 1379, VOL. 53] 



used in calculaling the potential of a mass from its density, we 

 shall have by Poisson's theorem 



vHd^/dt) = 4itd^:B/dt-, 



V"(d^jdt) = - 4Tc"*v-35, 



d%ldt = - 4irf-2-D . . 

 From this, with (4), and the general equation 

 d^/dt + 4irc-"-D -h vV = o, 



or by (5), 

 whence 



(9) 



we see that during the discharge of the system (A, B) the 

 electrostatic force - vV vanishes throughout all space, while its 

 place is taken by a precisely equal electrodynamic force 

 - d^/dt. 



This electrodynamic force remains unchanged at every point 

 until the passage of the waves, after which the electrostatic 

 force, the electrodynamic force, and the displacement, have the 

 permanent value zero. 



If we write Curl for the differentiating vector operator which 

 Maxwell calls by that name, equations (8) may be put in the 

 form 



4ir!j) = Curl Curl (iu), 

 whence 



d-l)/dt = (47r)-' Curl Curl (idu/dt). 



PVom d^/dt we may calculate the magnetic induction SB by 

 an operation which is the inverse of (4ir)"^ Curl. We have 

 therefore 



33 = Curl (idu/dt), 

 or 



SB = [r-3F'(/ - cr) + cr'^FV - cr)](yi - zj). 

 The magnetic induction is therefore zero except in the waves. 



Equations (4) and (9) give the value of d^jdt as function of 

 (/ and r). By integration, we may find the value of 91, Max- 

 well's " vector potential." This will be of the form of the 

 second member of (4) multipled by - c~^, if we should give 

 each F one accent less, and for an unaccented F should write 

 F„ to denote the primitive of F which vanishes for the argu- 

 ment 00 . 



That which seems most worthy of notice is that although 

 simultaneously with the discharge of the system (A, B) the 

 values of what we call the electric potential, the electrodynamic 

 force of induction, and the " vector potential," are changed 

 throughout all space, this does not appear connected with any 

 physical change outside of the waves, which advance with the 

 velocity of light. 



If we now suppose that there is a second pair of charged 

 spheres (t, d), as in the original problem, the discharge of this 

 pair will evidently occur when the relaxation of electrical dis- 

 placement reaches it. The time between the discharges is, 

 therefore, by Maxwell's theory, the time required for light to pass 

 from one pair to the other. 



It may also be interesting to observe that in the axis of x» on 

 both sides of the origin, XP = ''% and equation (4) reduces to 



4ir35 = [2r-^Y{t - cr) + 2cr-^F'{t - cr)]i. 

 Here, therefore, the oscillations are normal to the wave-surfaces* 

 This might seem to imply that plane ^nes of normal oscilla- 

 tions may be propagated, since we are accustomed to regard a 

 part of an infinite sphere as equivalent to a part of an infinite 

 plane. Of course, such a result would be contrary to Maxwell's 

 theory. The paradox is explained if we consider that the parts 

 of the wave-motion, expressed by F and F', diminish more 

 rapidly than those expressed by F", so that it is unsafe to take 

 the displacements in the axis of x as approximately representing 

 those at a moderate distance from it. In fact, if we consider the 

 displacements not merely in the axis of x» l>ut within a cylinder 

 about that axis, and follow the waves to an infinite distance 

 from the origin, we find no approximation to what is usually 

 meant by plane waves with normal oscillations. 



J. WlLLARD GiBBS. 



New Haven, Conn., March 12. 



An Unusual Solar-Halo. 

 On March 17, at Gottingen, a curious solar halo was observed 

 by a friend and myself towards the time of sunset. The weather 

 that day had been l>eautifully fine, but towards 5h. p.m. (Mean 

 P2uropean Time) thin light clouds began to form, which coveretl 

 the heavens with a thin white raiment. When the sun was 



