May 9, 1889] 



NATURE 



33 



-Y — -r 



dx dy dz 



If we now assume that during the passage of a disturbance of 

 the kind Q = .8 = constant, and they must be if 



'^^ + '^^ + '^ - o, 

 dx dy dz 

 we get — 



dt dy dy dy 



and consequently — 



^ (R)8 - Q-y) = V '/ (PQ + a^;. 

 dt dy 



Comparing this with Sir William Thomson's equation (34) we 

 see that — 



/= - 5^^-'^, xzav{uv) = PQ + aj3). 

 Similarly, by calculating 



^^(PQ + «)3) = Q'^P+ 35« 

 at at at 



= _ QV /')' + 8V '^,'^ = + V f (R8 - Q7), 

 dy dy dy 



we reproduce Sir William Thomson's equation (49), if V- = |R-, 

 i.e. be I of the average square of velocity of turbulency. This 

 V is the velocity of propagation of the disturbance. 



If we wish to identify this laminar motion of Sir William 

 Thomson's with a simple wave propagation such as light consists 

 of, we must take Q = 3 = constant, and then the two equations 

 are satisfied by — 



dV^ 

 dt 



and 



= -X'^y and ^ = 

 dy dt 



V 



dV 



dR _ Y "'« 



da 



dt 



1 iia. -.T dR 

 and = V , 



dy dt dy 



no matter what Q and j8 are, and these are, of course. Maxwell's 

 equations. There is nothing to settle which is the electric and 

 which the magnetic disturbance, nor even which,/ or x3az'(M^'), is 

 proportional to R^S - Q7, and which to PQ + o3, but the 

 consideration that electric currents and electrification are pos- 

 sible while magnetic currents do not exist, will probably decide 

 a question of this kind. In Maxwell's simplest wave, P and 7 only 

 exist, and in this case, as I have assumed above, xzav{tiv) would 

 correspond to electric, and / to magnetic disturbance. In Sir 

 William Thomson's representation, xzav[uv) is of the nature of 

 a twist, andy"of a flow, contrary to the usual notion that mag- 

 netic force is twisty. However, a flow cannot take place outwards 

 continuously from a body, so that there seems a reasonableness 

 in likening electrification to a twist. The fact that magnetism 

 in matter rotates the plane of polarization sometimes to one side 

 and sometimes to the other does not prove conclusively that it is 

 a rotation : a flow might confer that property on matter. 



In order to include the general case of a variable state, an 

 interpretation of X, Y, Z, is required. Where no matter is 

 present, we must assume — 



- X = '^^L + i '^(P2 + qi + R2 + a2 + ^2 + ^is^ szc, 

 dx ax 



and in the steady state — 



A = - i(P- + Q' + R- + a^ + )8= + -f-y 



When the state is not steady, we have, if 



d? 



dx 



- + 



dz 



da 



+ 



+ 



dy _ 



du 



X - «J^ = + p^ + am + 

 dt 



dx ' dy ' dz 

 djQy - R18) 

 dt V 



, &c. 



We must assume that pi has no longer the above value ; but by 

 differentiating the first of these with respect to x, the second 

 with respect to>', and the third with respect to z, we get — 



a^X ^ rt'Y ^ ^Z _ d/dtt ^ dz! _j_ dw\ 

 dx dy dz dt\ dx dy dz I 



= + 



{'•■ 



+ p J + Q^ + R'J 

 dx dy dz 



. I d^. 



) + {„.' 



, dm , ad»i , dm\ 



dx 



= V2^^<P' + Q'+ R'^ + «^+S^ + 7-), 



which is satisfied by « = o, »; = o ; or if ^ and m exist, by — 



.de 



de 



t^ + ?~ + Q'^ + R 



and 



and 

 and 



\c!X- 



111- + a 



dx 

 dm 



dy 



^ adm 

 dx dy 



+ r 



dz 

 dm 



du 



dx 



dv , 

 dy ^ 



dy' dz- 



dw d^p, 



dz dx- 



dz 



dy, 



dy' 



dz- 



1 



V3 



;>- 



+ Q- + R2 + a- + )3- + 7") = o. 



which means that energy is propagated with a velocity = V, 

 and so the assumed relations connecting P, Q, R, and a, /3, 7, 

 mean little more than that the initial state is stable. 



This, I think, shows that, so far, the ether may be a turbulent 

 liquid. 



If we compare the dimensions of the quantities involved in 

 the theory of motion of a turbulent liquid with those in the 

 electro-magnetic theory, we find it convenient to put these latter 

 dimensions into the following forms, as they are the same on the 

 electro-magnetic and electrostatic systems. 



Calling [K-^i] = [V"'], and density [p] = [ML"'], we can 

 write — 



Electric displacement = [K^p'V] 

 Electric force = [K-ip^V] 



Magnetic displacement = [ju^piV] 

 Magnetic force = [/u-ip^V]. 



It is at once evident that the products of the force and dis- 

 placement are, in each case, of the dimensions of the P'^, Q*, 

 R-, o^, ff-, 7-, involved in the theory I have already given. 



I think it would be well, perhaps, to introduce some new 

 quantities of zero dimensions to define the polarization of the 

 medium. It seems likely that the velocity involved in P must 

 depend on how intensely the turbulency is polarized, and could 

 therefore be measured by a quantity of zero dimensions multi- 

 plied by a measure of the turbulency. This measure would be 

 p^V, so that, for electrostatic energy — 



P- = VoHpY-), 



and for electro-magnetic — 



«o"(pV-), 



where P,, and a,, were of zero dimensions. 



In order to introduce the eff"ect of alterations in material, we 

 may put these in the form — 



while the magnetic displacement will be — 



= ./" • «"' 

 \^ K 



and the magnetic force will be — 



_ / p "0 

 V K ' M 

 If we call the six quantities iP, v-, vr, vw, wu, uv, a, h, f, 

 /, g, li, they are connected with the six quantities P, Q, R, 

 o, j8, 7, and the three undisturbed values of iP = A, z>- = B, 

 w- = C, by the equations — 



rt = A -f P2 -t- aS &c., /= QR H- /37, &c. 



In order to examine how these are related, take an ellipsoid 

 defined by — 



ax- + by- -I- cz- ■\- 2fyz + 2gzx + 2hxy = d. 



