36-38] The Method of General Dynamics 37 



(p. 25) we may suppose the former limits represented by the parallelepiped a, 

 in which case the latter set will be represented t by the parallelepiped /3. 

 These parallelepipeds have been shewn to be equal in size although in the 

 present case the orientations are not the same. This proves that the size of 

 the element of volume of generalised space occupied by the series of systems 

 now under consideration is unaltered by a collision of molecule A with the 

 boundary, and hence that the density of the fluid may be supposed to remain 

 unaltered. 



The case of a collision between a pair of molecules may be treated in 

 the same way. If the molecules are A and B, all the coordinates remain 

 unaltered except u a , v a ,w a , u b , v b , w b , and the result follows at once from 

 equation (9) if .we change the notation so as to replace u, v, w by a a , v a , w a 

 and u', v', w' by u b , v b ,w b . 



Hence, examining the motion of any small element of fluid in our general- 

 ised space, we have proved that the density of this element remains unchanged 

 by steady motion and by collisions, i.e., remains unchanged throughout the 

 whole motion of the gas. It follows that if the whole generalised space is 

 filled with fluid initially homogeneous, then this fluid will remain homo- 

 geneous throughout the entire motion. 



38. It has already been remarked that the stream-lines along which the 

 fluid moves are permanently fixed in the generalised space. This fact, 

 combined with the result just proved, shews that the motion of the fluid we 

 are discussing is a " steady-motion " in the hydrodynamical sense. 



One further feature of this motion must be noticed. If we denote the 

 total kinetic energy of any system by E, so that 



2E=m(u a 2 + v a 2 + w a 2 +u l ? + ...) (57), 



it is clear that E remains constant throughout the whole length of any 

 stream-line. When E is a constant, equation (57), regarded as a relation 

 between the Cartesian coordinates of a point in the generalised space, expresses 

 that the point lies on a certain locus (of dimensions QN 1) in this space. 

 It follows, then, that the motion of any element of the fluid is confined to 

 that member of the family of loci E = constant, in which it started. 



To obtain some idea of the disposition of this family of loci in our 

 generalised space, we notice that 2E/m is the square of the perpendicular 

 distance from 



u a = v a = w a = u b = . . . = 0, 



or, what is the same thing, from E=0. Hence the loci enclose one another, 

 being in fact a system of tubular surfaces of which the cross-sections 

 are spherical loci of 3N dimensions. The tubes do not extend to infinity 

 along their length. For we pass along a generator of a tube by varying 

 #a> 2/a, z a , #&...etc., and none of these coordinates can become infinite, 



