42 Mb maxwell, ON FARADAY'S LINES OF FORCE. 



(33) Suppose that in a uniform medium the motion of the fluid is everywhere parallel 

 to one plane, then the surfaces of equal pressure will be perpendicular to this plane. If we 

 take two parallel planes at a distance equal to k from each other, we can divide the space 

 between these planes into unit tubes by means of cylindric surfaces perpendicular to the planes, 

 and these together with the surfaces of equal pressure will divide the space into cells of which 

 the length is equal to the breadth. For if h be the distance between consecutive surfaces of 

 equal pressure and s the section of the unit tube, we have by (IS) s = kh. 



But s is the product of the breadth and depth ; but the depth is k, therefore the breadth 

 is h and equal to the length. 



If two systems of plane curves cut each other at right angles so as to divide the plane 

 into little areas of which the length and breadth are equal, then by taking another plane at 

 distance k from the first and erecting cylindric surfaces on the plane curves as bases, a system 

 of cells will be formed which will satisfy the conditions whether we suppose the fluid to run 

 along the first set of cutting lines or the second *. 



Application of the Idea of Lines of Force. 



I have now to shew how the idea of lines of fluid motion as described above may be 

 modified so as to be applicable to the sciences of statical electricity, permanent magnetism, 

 magnetism of induction, and uniform galvanic currents, reserving the laws of electro-magnetism 

 for special consideration. 



I shall assume that the phenomena of statical electricity have been already explained by 

 the mutual action of two opposite kinds of matter. If we consider one of these as positive 

 electricity and the other as negative, then any two particles of electricity repel one another with 

 a force which is measured by the product of the masses of the particles divided by the square 

 of their distance. 



Now we found in (18) that the velocity of our imaginary fluid due to a source 5" at a distance 

 r varies inversely as r°. Let us see what will be the efffect of substituting such a source for 

 every particle of positive electricity. The velocity due to each source would be proportional to 

 the attraction due to the corresponding particle, and the resultant velocity due to all the 

 sources would be proportional to the resultant attraction of all the particles. Now we may 

 find the resultant pressure at any point by adding the pressures due to the given sources, and 

 therefore we may find the resultant velocity in a given direction from the rate of decrease 

 of pressure in that direction, and this will be proportional to the resultant attraction of the 

 particles resolved in that direction. 



Since the resultant attraction in the electrical problem is proportional to the decrease of 

 pressure in the imaginary problem, and since we may select any values for the constants in 

 the imaginary problem, we may assume that the resultant attraction in any direction is nume- 

 rically equal to the decrease of pressure in that direction, or 



ao) 



See Cambridge and Dublin Mathematical Journal, Vol. III. p. 286. 



