OF PHYSICAL QUANTITIES. 261 



Vectors which are referred to unit of length I shall call Forces, using the 

 word in a somewhat generalized sense, as we shall see. The operation of taking 

 the integral of the resolved part of a force in the direction of a line for 

 every element of that line, has always a physical meaning. In certain cases the 

 result of the integration is independent of the path of the line between its 

 extremities. The result is then called a Potential. 



Vectors which are referred to unit of area I shall call Fluxes. The opera- 

 tion of taking the integral of the resolved part of a flux perpendicular to a 

 surface for every element of the surface has always a physical meaning. In 

 certain cases the result of the integration over a closed surface is independent, 

 within certain restrictions, of the position of the surface. The result then 

 expresses the Quantity of some kind of matter, either existing within the surface, 

 or flowing out of it, according to the physical nature of the flux. 



In many physical cases, the force and the flux are always in the same 

 direction, and proportional to each other. The one is therefore used as the 

 measure of the other, their symbols degenerate into one, and their ideas become 

 confounded together. One of the most important mathematical results of the 

 discovery of substances having different physical properties in different directions, 

 has been to enable us to distinguish between the force and the flux, by letting 

 us see that their directions may be different. 



Thus, in the ordinary theory of fluids, in which the only motion considered 

 is that which we can directly perceive, we may define the velocity equally well 

 in two different ways. We may define it with reference to unit of length, as 

 the number of such units described by a particle in unit of time ; or we may 

 define it with reference to unit of area, as the volume of the fluid which 

 passes through unit of area in unit of time. If defined in the first way, it 

 belongs to the category of forces ; if defined in the second way, to the category 

 of fluxes. 



But if we endeavour to develope a more complete theory of fluids, which 

 shall take into account the facts of diffusion, where one fluid has a different 

 velocity from another in the same place ; or if we accept the doctrine, that the 

 molecules of a fluid, in virtue of the heat of the substance, are in a state of 

 agitation ; then though we may give a definition of the velocity of a single mole- 

 cule with reference to unit of length, we cannot do so for the fluid ; and the only 

 way we have of defining the motion of the fluid is by considering it as a flux, 

 and measuring it by the mass of the fluid which flows through unit of area. 



