,, MATHEMATICAL CLASSIFICATION 



Thw distinction in still more necessary when we come to heat and electricity. 

 The flux of beat or of electricity cannot be even thought of in any way except 

 M the quantity which flows through a given area in a given time. To form a 

 conception of the velocity, properly so called, of either agent, would require 

 us to conceive heat or electricity as a continuous substance having a known 



density. 



We must therefore consider these quantities as fluxes. The forces corre- 

 sponding to them are, in the case of heat, the rate of variation of temperature. 

 and, in the case of electricity, the rate of variation of potential. 



I have said enough to point out the distinction between forces and fluxes. 

 In statical electricity the resultant force at a point is the rate of variation of 

 potential, and the flux is a quantity, hitherto confounded with the force, which 

 I have called the electric displacement. 



In magnetism the resultant force is also the rate of variation of the 

 potential, and the flux is what Faraday calls the magnetic induction, and is 

 measured, as Thomson has shewn, by the force on a unit pole placed in a narrow 

 crevasse cut perpendicular to the direction of magnetization of the magnet. I 

 shall not detain the Society with the explanation of these quantities, but I 

 must briefly state the nature of a ratio of a force to a flux in its most general 

 form. 



When one vector is a function of another vector, the ratio of the first to 

 the second is, in general, a quaternion which is a function of the second 

 vector. 



When, if the second vector varies in magnitude only, the first is always 

 proportional to it, and remains constant in direction, we have the important 

 case of the function being a linear one. The first vector is then said to be a 

 linear and vector function of the second. If a, /3, y are the Cartesian compo- 

 nents of the first vector, and a, b, c those of the second, then 



where the coefficients p, q, r are constants. When the p'a are equal to the 

 corresponding 5*8, the function is said to be self-conjugate. It may then be 

 represented geometrically as the relation between the radius vector from tin- 

 centre of an ellipsoid, and the perpendicular on the tangent plane. 



