390 MATHEMATICAL APPENDICES 19* 



where the upper limit of the integral is 



= 0/W. 



To determine the value of M* a numerical evaluation of the above 

 transcendental integral is necessary, which can be carried out by 

 means of different rapidly converging series ; and we find 



= 1-0875, 

 and therefore 



= 1-0875 TT= 0-9640. 



Thus the mean probable value is greater than the most probable 

 W, whereas it is less than the other two mean values, so that the 

 four special values of the molecular speed arranged in order of 

 magnitude form the series l 



W < < O < G. 



To exhibit these relationships more clearly some numerical 

 examples calculated from these formulae have been given in 28. 



We need not here examine more closely the more general case 

 wherein there is also a translatory motion of the gas as a whole 

 in addition to its molecular motion, since the formulas for the 

 mean energy and the pressure have been already deduced in a 

 more general investigation in 7*. 



20*. Mixed Gases 



The foregoing investigation may be easily extended to the more 

 general case in which the gaseous medium considered consists of 

 a mixture of different kinds of molecules. The calculation is 

 of exactly the same character, the only difference being that the 

 number of equations of condition is increased. 



In order to avoid unnecessary complications let us assume that 

 the gaseous mixture is not in a state of motion as a whole, but, 

 apart from the molecular heat-motions, is at rest and in equili- 

 brium ; then for each kind of molecules three equations hold good 

 of the form 



where m lt ra 2 , . . . denote the masses of the different kinds of 

 1 [These means are very approximately as 80 : 87 : 90 : 98 T.]. 



