448 MATHEMATICAL APPENDICES 47* 



which is carried over in the opposite direction, when v' 2 is put 

 for v f . 



In this formula the function B, which expresses the probability 

 of a collision, is, strictly speaking, not only dependent on the 

 velocity o>, but also varies with place, that is with r, since the 

 gas is not in the same state of motion at all points. We may, 

 however, neglect this variation, and take B to have the same value 

 everywhere if we retain the assumption that the velocities v and v f 

 of the forward motion may be looked upon as vanishingly small 

 in comparison with the mean speed 1 of the heat-motion. 



In this approximation we can further neglect the square of v f 

 in comparison with <o 2 , and put 



e~* = e~ kma> \l + Qkmuv' sin s cos 0), 



by which the integrations become partly practicable, and we 

 obtain a development in a series proceeding by powers of w/B = I, 

 which we may limit to its first terms ; we then have 



Q l = (yv + ^rj dv/dx)dy dz, 

 and also 



62 = (yv i? dv/dx)dydz, 

 wherein 



15?r 3 



The last formula gives the value of the coefficient of friction of 

 the gas. 



48*. Investigation and Development of the 

 Formula for the Coefficient of Viscosity 



This formula is as little integrable as Boltzmann's; but 

 even thus it is not difficult to grasp its meaning and deduce 

 from it the laws of internal friction, just as from the formulae 

 first obtained. 



We first of all easily see that by this calculation too the 

 value of the coefficient of viscosity is proportional to the square 

 root of the absolute temperature. For if in the formula for r/, 

 which may be written 



