232 NOTES. 



which therefore necessarily consist each of an immense number of 

 molecules. 



In order then to establish the fundamental equations in a manner 

 free from special hypothesis as to the structure of matter we adopt the 

 following conventions. 



We suppose the * elements above spoken of to be such that each of 

 their dimensions is a large multiple of the average distance (d, say,) 

 between the centres of inertia of neighbouring molecules, and also of 

 the average distance (8, say,) beyond which the direct action of one 

 molecule on another becomes insensible*. The latter proviso is neces 

 sary in order that the mutual forces exerted between adjacent elements 

 shall be sensibly proportional to the surfaces across which they act. 

 Observation shews that we may suppose the dimensions of an element 

 to be at the same time so small that the average properties of the 

 constituent molecules vary regularly and continuously as we pass from 

 one element to another. We shall, in what follows, understand by the 

 word particle or element a portion of matter whose dimensions lie 

 within the limits here indicated. The properties of an element sur 

 rounding any point P may then be treated as continuous functions of 

 the position of P. 



The density at any point of the fluid is now to be denned as the 

 ratio of the mass to the volume of an elementary portion surrounding 

 that point. 



The * velocity at a point is denned as the velocity of the centre of 

 inertia of an elementary portion taken about that point. This is of 

 course quite distinct from the velocities of the individual molecules, 

 which may, and in all probability do, vary quite irregularly from one 

 molecule to another. 



By the flux across an ideal surface situate at any point of the 

 fluid we shall understand the mass of matter which in unit time crosses 

 unit area of the surface, from one side to the other. Matter crossing 

 in the direction opposite to that in which the flux is estimated is here 

 reckoned as negative. 



The flux across any surface at any point is equal to the product of 

 the density (p) into the velocity (&amp;lt;/, say,) estimated in the direction of the 

 normal to the surface. This is sufficiently obvious if the fluid be 

 supposed continuous, or even if it be molecular, provided that in the 



* It is supposed that in gases d is large compared with 5 ; the reverse is 

 probably the case in liquids and solids. 



