534 Mr. S. H. Burbuiy on 



On the Definition of Stream at a Point. 



13. Evidently v.- hem we speak o£ the stream at a point P we 

 have in our mind many molecules near P. And we do not 

 regard molecules very distant from P. The way to express 

 the idea is to weight the molecules (so to speak) according to 

 their nearness to P. If m be the mass, u the oc velocitv of a 

 molecule, let ^=^f .mit be the stream in x at P, the sum- 

 mation including all molecules, and / being a function of r, 

 the distance from P which, when r exceeds a certain minimum, 

 diminishes very rapidly with increasing o\ For instance, 

 when r > c 



J ,,.^ ' 



where A and k are constants. Let rj, ? have corresponding 

 meaning for the component velocities v and ic. 



Suppose_, then, there is a molecule at P having velocities 

 w, Vy ic. I should then define ^ (u^ + vrj + ic^) to be the energy 

 of the stream motion for P, and (f) = ^'L{u^-^V7} + w^) the 

 whole stream energy. (^ is of course zero under assumption A.- 

 But if assumption A be not made, it is on average positive 

 for repulsive, negative for attractive, forces. That indicates 

 that the molecules having finite dimensions choose in either 

 case the least resisted motion. When the forces are repulsive 

 the energy of the relative motion of neighbouring molecules 

 is less than, according to assumption A, it should be — and the 

 difference is the stream energy. For attractive forces the 

 enero'Y of the relative motion of a mole( ule and its immediate 

 neighbours is greater than according to assumption A it 



at 



should be. In steady motion <^ must be constant or -^ =0. 



That is 



i2(^ 



dt dt 



dt 



-\-i 



)=». 



which can be reduced on average to 



' •(■2-|-«;f) 



For steady motion we should, accordino- to Profe^^or Brvan^s 



dr- 



1 ^"^"^ A 



canon, make -~ = 0. 



