e -km(w-y)2^ w dw> 



Molecular Velocities amongst the Molecules of a Fluid. 169 



/I CO f* tt> /*co 



1 = B \ € - km( - u -«') 2 . du = B \ 6 - km ^-^\ dv = B I e.-^Cw-v) 2 . dw. 



J— CO J— CO J — X 



Hence 



1=B \/l < 15 > 



Again, since a, b, c are the mean values of the component 

 velocities, 



a = B| e- km ^-^\udu, 



*) — CO 



6 = bT e-^'-^.y^ 



»/ -co 



J CO 

 -c 



These give, by help of (15), 



u = a ; ft=b ; y = c. 

 Equation (14) may therefore be written 



F(W, V, iv)=l—ye-km((u-a)2 + (v-b^+(w-c)^ . dll dv dw. (16) 



Let us return now to equation (10), and substitute in it the 

 value of a, ft, and 7. Then 



j (a Sa + 6 86 + c Sc)2m„=0. 



0r 8T=0; (17) 



where 



T=ijV + & 2 + c 2 )2»v 



The quantity T is evidently the molar kinetic energy of the 

 fluid. The condition ST = expresses that this kinetic energy 

 is a maximum, or a minimum, or a maximum-minimum. 



Hence the theorem : — If the molar kinetic energy of a fluid 

 fulfils a maximum, or a minimum, or a maximum-minimum 

 condition, then the distribution of molecular (linear) velocities 

 follows a law of the same form as that of Maxwell for gases. 



Gordon's College, Aberdeen, 

 January 1888. 



