MATHEMATICAL CLASSIFICATION 



Thus, if we take the mass and the square of the velocity as the factors, 

 M done in the ordinary definitions of vis-viva or kinetic energy, both factors 

 are acalar, though one of them, the square of the velocity, has no distinct physical 



meaning. 



Another division into apparently scalar factors is that into volume and 

 hydrostatic pressure, though here we must consider the volume, not in itself, 

 but as a quantity subject to increase and diminution, and this change of volume 

 can only occur at the surface, and is due to a variation of the surface in the 

 direction of the normal, so that it is not a scalar but a vector quantity. The pressure 

 also, though the abstract conception of a hydrostatic pressure is scalar, must be 

 conceived as applied at a surface, and thus becomes a directed quantity or vector. 



The division of the energy into vector factors affords results always capable 

 of satisfactory interpretation. Of the two factors, one is conceived as a tendency 

 towards a certain change, and the other as that change itself. 



Thus the elementary definition of Work regards it as the product of a force 

 into the distance through which its point of application moves, resolved in the 

 direction of the force. In the language of Quaternions, it is the scalar part of 

 the product of the force and the displacement. 



These two vectors, the force and the displacement, may be regarded as 

 types of many other pairs of vectors, the products of which have for their 

 scalar part some form of energy. 



Thus, instead of dividing kinetic energy into the factors " mass " and " square 

 of velocity," the latter of which has no meaning, we may divide it into "mo- 

 mentum " and " velocity," two vectors which, in the dynamics of a particle, are 

 in the same direction, but, in generalized dynamics, may be in different direc- 

 tions, so that in taking their product we must remember the rule for finding 

 the scalar part of it. 



But it is when we have to deal with continuous bodies, and quantities 

 distributed in space, that the general principle of the division of energy into 

 two factors is most clearly seen. 



When we regard energy as residing intrinsically in a body, we may measure 

 its intensity by the amount contained in unit of volume. This is, of course, a 

 scalar quantity. 



Of the factors which compose it, one is referred to unit of length, and the 

 other to unit of area. This gives what I regard as a very important distinction 

 among vector quantities. 



