NOTES. 233 



latter case we assume the velocities of all the molecules within an 

 element to agree in magnitude and direction. To prove the statement 

 when the velocity is supposed to vary quite irregularly from one 

 molecule to another, we have only to suppose the molecules contained 

 in an elementary space to be grouped according to their velocities, so 

 that the velocities of all the members of any one group shall be sensibly 

 the same in magnitude and direction. Let p lt q x be the values of the 

 density, and of the velocity in the direction of the normal to the given 

 surface, corresponding to the first group alone, p 2 , q 2 the corresponding 

 values for the second group, and so on. The part of the flux due to the 

 first group is p^ lt that due to the second is p 2 q^ and so on, so that the 

 total flux in the given direction is 



which is, by the definition of the symbols, = pg. 



Hence the flux across any portion of a surface every point of which 

 moves with the fluid is zero. 



The foregoing considerations and definitions apply alike to solids 

 and to fluids. But if we attempt to follow the motion of an element 

 we are met by a difficulty peculiar to our present subject. &quot;We cannot 

 assume that the molecules which at any instant constitute an element 

 continue to form a compact group throughout the motion. On the 

 contrary the phenomena of diffusion shew that such an association of 

 molecules is gradually disorganized, some molecules being continually 

 detached from the main body, whilst others find their way into it from 

 without. Thus although the matter included by a small closed surface 

 moving with the fluid is constant in amount, its composition is contin 

 ually changing. It is true that in liquids this process is exceedingly 

 slow, and might fairly be neglected when regarded from our present 

 point of view. In the case of gases it must however be taken into 

 account, in consequence of the much greater mobility of their molecules. 



The phrase * path of a particle, often used in the text, must there 

 fore now be understood to mean the path of a geometrical point which 

 moves always with the velocity of the fluid where it happens to be. In 

 the case of a liquid this will represent with considerable accuracy the 

 path of the centre of inertia of a definite portion of matter. 



The effect of the foregoing definitions is to replace the original 

 (molecular) fluid by a model, made of an ideal continuous substance, in 

 which only the main features of the motion are preserved. The corre 

 spondence of the model to the original is however as yet merely 



