ON FARADAY'S LINES OF FORCE. 175 



(33) Suppose that in a uniform medium the motion of the fluid is every- 

 where 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 (13) 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 elec- 

 tricity, 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 S at a distance r varies inversely as r*. Let us see what will be the 

 effect 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 



* See Cambridge and Dublin Mathematical Journal, Vol. HI. p. 286. 



