432 Prof. Challis on the Hydrodynamical Theory of the 



dt) 



currents be in the same direction, -j- is a positive quantity, and 



p increases with r, so that the action tends to diminish the in- 

 terval r— h, or is attractive. But if V, and V, have different 



signs, or the currents be in contrary directions, ~ is negative, 



and the action is repulsive. It is plain that the argument ap- 

 plies to the action of either rheophore on the other. 



57. Since in the foregoing reasoning the lengths of the rheo- 

 phores were not involved, we may draw from it the important 

 corollary, that, if two elements of parallel currents be so situated 

 that the straight line joining their middle points is perpendicu- 

 lar to their axes, the mutual action between the elements varies 

 inversely as the square of the interval between the axes. It is to 

 be noticed that this law has thus been independently derived by 

 argument from the fundamental hypotheses. 



58. " We may always replace a rectilinear current by any 

 sinuous current in the same general direction, provided it is 

 very little distant from the first " (Jamin, p. 200). This is 

 another of the facts, or laws, which, as forming the basis of 

 Ampere's theory, it is proposed to account for by the hydro- 

 dynamical theory. The following is, I think, the appropriate 

 argument in this instance. 



According to the principles adduced in arts. 11 and 34, in 

 order that any element of a galvanic current may transfer a 

 portion of fluid across a plane perpendicular to its direction, it 

 is necessary that there should be a stress, of the nature of 

 hydrostatic pressure, to counteract the resistance arising from 

 the inertia of the surrounding fluid, and that the total current 

 of which the element is a part should circulate by means of a 

 rheophore. The first condition is fulfilled by the force which 

 has its origin at the galvanic battery, and the other by a closed 

 conductor. Now this stress, like any hydrostatical force, is 

 resolvable into components in rectangular directions, and in the 

 instance of the sinuous rheophore may be resolved at each 

 point in the mean direction of the rheophore, and in two direc- 

 tions at right angles to this. The mode of action of the stress 

 may be conceived of by reference to the proof of the equality 

 of hydrostatic pressure in all directions usually given in elemen- 

 tary treatises on Hydrostatics. A closed vessel of any form is 

 supposed to be completely filled with incompressible fluid j and 

 by means of a piston inserted into a side of the vessel a pres- 

 sure is produced which by the intervention of the fluid is 

 equally exerted at all points of its interior and of the enveloping 

 surface, so that if at any other part of the containing vessel a 

 movable piston were inserted, it would be put in motion by the 



