18* MAXWELL'S LAW 387 



whose direction is given by the angles s and with reference to 

 a fixed axis, such that 



u = w cos s, 



v = w sin s cos 0, 



w = a> sin s sin </>. 



In order not unnecessarily to complicate the calculation, 

 which I wish to carry out without the limiting assumption of a 

 state of rest, I take the position" of the coordinate system, which 

 so far has been left arbitrary, such that the axis of u, which is 

 also that of the polar system of coordinates, coincides with the 

 direction of the absolute velocity of translation of the whole 

 system 



o= v/(a 2 + 6 2 + c 2 ). 



Then in the former formulae o enters instead of a, while b and 

 c vanish altogether. Since, further, the element of volume is now 

 given by the expression w 2 ^w sin s ds d$, we have, instead of the 

 first formula of 16*, the new one 



n = N^km^e-^^-^^^^^da, sin s ds d<j>, 



and this gives the number of molecules which out of every N move 

 with the velocity w in the direction given by s and </>. 



From this we obtain by integration the whole number v of 

 all those which move with a speed lying between w and w + da, 

 viz. 



ds sin s er^-^ cos + a > 



e ~ * m( ~ 



With the special assumption that there is no translatory 

 motion, i.e. that the gas is at rest as a whole or o = 0, this 

 formula, found by Maxwell, becomes 



This new formula differs from the former formula in 16* by 

 an important circumstance ; while the latter showed a continuous 

 diminution of the probability as the values of the components 

 increased, this has a maximum which occurs for the value 



= 1 

 or 



w = (km)-*= W. 



c c 2 



