Feb. 21, 1889] 



NATURE 



403 



dt 

 dt 



dt 



dt 



A '^^ : 

 dt 



dZ 

 dy ' 



^. 



dz 

 dY 

 dx 

 dU 

 dz 

 dN 

 d.v 

 dL 

 dy ■ 



dz I 

 rfZ ( 

 dx I 

 dX. I 

 dy } 

 dH 

 'dy 

 dL 

 dz 

 dJA 

 dx 



(0 



(2) 



triginally, and therefore always, the followins 

 -atisfied : 



conditions must 



dL 

 dx 



dM 

 dy 



^=°. .(3) 



^N-=o,and'^X^^Y 

 dz dx dy 



The electric energy contained in a portion of ether of volume 



T IS — 



the magnetic energy is — 



u<- 



M' 



N2)r/T 



the integration extending all through the volume. The total 

 energy is the sum of both these portions. 



These expressions form the essential ingredients of Maxwell's 

 theory as it relates to the ether. Maxwell arrived at them by 

 forsakmg action at a distance, and by accummodating the ether 

 with the properties of a highly dielectric medium. One can 

 also get the same equations in another way. But hitherto no 

 direct proof of the validity of these equations has been afforded 

 by experience. It appears most logical, therefore, to regard 

 them independently of any way in which they may have been 

 arrived at, to consider them as hypothetical assumptions, and to 

 let their probability depend upon the very great number of 

 legitimate conclusions which they embrace. Taking this point of 

 view, one can do without a series of auxiliary ideas, which render 

 the understanding of Maxwell's theory more difficult, even if on 

 no other ground than that, so soon as one finally excludes the 

 hypothesis of immediate action at a distance, these notions 

 possess no meaning. 



Multiply equations (i) by L, M, N, and (2) by X, Y, Z ; add 

 the equations together, and integrate over the whole space, 

 whose volume element is dr, and whose surface element is dS ; 

 we gel — 



4:J«^ 



Y - MZ) X + (LZ - NX) M + (MX - LY)^]dS, 



where A, ^, v are the direction-cosines of the normal to the surface. 



This equation shows that the amount by which the energy of 

 the space has increased can be regarded as having entered 

 through its walls. The quantity entering through any single 

 element of surface is equal to the product of the components of 

 the electric and magnetic forces which belong to that element, 

 multiplied by the sine of the angle between them, and divided 

 by 4T.A.. On this result it is well known that Prof. Poynting 

 has founded a remarkable theory on the transfer of energy in 

 the electro-magnetic field. 



For the purpose of solving the equation, we limit ourselves to 

 the special but important case where the distribution of the 

 electric force is symmetrical about the axis of z, and hence that 

 this force is absent at every point of the meridian planes in- 

 tersecting in the axis ofz, and only depends on the ;: co-ordinate of 

 a point, and on its distance, r = Vjc- V j^, from the 2 axis. We 

 will denote the electric force in the direction of r, namely, 

 X + Y-*- , by K ; and the component of the magnetic force 



which is normal to the meridian planes, viz. L-^ - M-'-,byP. 



r r •' 



We assert further that if n is any function of r, z, and t, which 

 satisfies the equation — 



our equations : — 



^ - r dr' 



R= -L'.% 

 r iiz 



p = A ./Q 



r dt' 

 N = o. 

 To prove this assertion we observe that 

 dr ^ _ d-n 

 dx dxdz 



J, r/r _ _ d-Ti 

 dy dydz 



I rf_ 

 r dr 



X = R 



(d\i\ ^ dju 

 \ dr) dx' 



dr 



XI = - 

 N = o. 



dr 



d-^TI 

 dydt' 



d^n 



dxdt' 



One has only to insert these expressions into equations (i), (2), 

 (3), to find equations (2) and (3) identically satisfied, and (l) also 

 if we have regard to the differential equation for n. 



It may be mentioned that also inversely, neglecting certain 

 practically unimportant limitations, every possible distribution of 

 electric force which is symmetrical to the axis of 2 can be repre- 

 sented in the above form, but it is not necessary for the sequel to 

 substantiate this assertion. 



The function Q is of importance. The lines in which the sur- 

 face of rotation Q = const, cut the meridian planes are the lines 

 of electric force ; the construction of the same for one meridian 

 plane furnishes at every instant an immediate presentation of the 

 force distribution. 



If we cut the shell between Q and Q -t- </Q by a surface of rota- 

 tion round the axis of z, the product of electric force and surface 

 which Maxwell calls the "'induction " is for every such surface 

 the same. If we arrange the system of surfaces Q = const, in 

 such a way that Q increases in arithmetic progression, the same 

 statement remains true when we compare the sections of the 

 different shells with one another. 



In the plane diagram which consists of sections of the meridian 

 plane with the equidistant surfaces Q ~ constant, the electric 

 force is inversely proportional to the normal distance of consecu- 

 tive lines Q = const, only for the case when points compared lie at 

 the same distance from the axis of c. In general the rule is that 

 the force is inversely proportional to the product of this distance 

 and of the co-ordinate r of the point considered. 



If we introduce polar co-ordinates p and d they will, be like 

 this. 



o„ /^ 



,'P 



The figure represents an electric oscillator at origin of co-ordinates .is 

 intended to be understood by Hertz. 



TAe Forces concerned in a Rectilinear Oscillation. 



Let E denote a quantity of electricity, and / a length ; let 



w = — be a reciprocal length, and « — _ a reciprocal time ; 



and let us put 



n = E - sin {mp - nt). 



