7l) PROFESSOR TAIT ON THE 



ment of other expressions which will presently occur, it may be well to note 

 that a mere inversion of the order of integration, in either part of the above 

 double integral, changes it into the other part. 



Otherwise : — we may reduce the whole to an immediately integrable form 

 by the use of polar co-ordinates ; putting 



v = r cos , v x = r sin 6 , 



and noting that the limits of r are to oo in both parts, while those of 

 6 are to tt/4 in the first part, and tt/4 to tt/2 in the second. [This trans- 

 formation, however, is not well adapted to the integrals which follow, with 

 reference to two sets of spheres, because h has not the same value in each set.] 

 14. Whatever method we adopt, the value of the expression is found to be 



\Z 8 j- ns * =2 \/£h 



and, as the mean speed is (§ 3) 



7TOS- 



JttIi 



we obtain Clerk-Maxwell's value of the mean path, above referred to, viz., 



1 



?17TS 2 J2' 



But (in illustration of the remarks at the end of § 11) we might have defined 

 the mean number of collisions per particle per second as 



vv v ' or as ^v -j-x ; &c, &c. 



z,(n v p v ) X{n v p v lv) 



The first, which expresses the ratio of the mean speed to the mean free path, 

 gives 



2 7T71S 2 



J^h 0-677' 



and the second, which is the reciprocal of the mean value of the time of 

 describing a free path, gives 



1 7TW.S 2 



Jh W65Q' 



The three values which we have adduced as examples bear to one another 

 the reciprocals of the ratios of the above-mentioned determinations of the 

 mean free path. 



