230 
Proceedings of tlie Royal Society of Edinburgh. [Sess. 
From the remainder we find 
YdB + Y(dv . q)B + Vd(qv . B) - dy . YqB + Vi dVqB^ 
= YdB + Y(dv . q)B + Y . d(qy • B) 
- Y(dv . q)B - Y . q(dv • B) + Vj . dYqBj^ 
= YdB + Y . d(qv . B) - Y . q(dy • B) 
- V . d(qv B) + Y . q(dv . B) + Vdq . vB 
= YdB + Ydq div B , (23) 
where the suffix restricts the action of \7 to the vector carrying the same 
suffix. 
Equation (21) then reduces to 
Ydjj^(curl E + B + div B) . .... (24) 
Now will have different values according to the different directions of 
the various sets of tubes ; hence (unless all the tubes are parallel) we 
may write 
curl E + B + q^ div B = 0 . .... (25) 
From this, since is the velocity of any set of tubes, unless all the 
sets have a common velocity, we must have 
divB = 0, (26) 
and thus 
-curlE = B ...... (27) 
We have now shown that the first four laws of the ordinary theory of 
electro-magnetism are consequences of the assumptions which have been 
made. It may be observed that whereas, in the proof of tlie first two laws 
(3) and (4), no departure of importance is made from the method of Recent 
Researches, the proof just given of the laws (26) and (27) is quite different 
from that adopted in that work. This is rendered necessary by the 
purpose of the present paper, which is not to deduce the properties of the 
tubes from the known laws of electro-magnetism, but to show that, given 
the tubes with the (essential) properties assigned to them by Sir J. J. 
Thomson, the laws of electro-magnetism follow. 
5. It remains to discuss the forces acting on the electric particles. 
Referring to the figure on p. 228, let B be a particle at the end of the 
tube B, C, D. Then the change in L due to the displacement of the end 
of the tube from B to A (introducing a new segment BA), is by (13) 
3L — 8r(E -t A q^nB) , ..... (28) 
since 
8d^ = AB= -8r, 
