346 Appendix A 



APPENDIX A. 



INTEGRALS INVOLVING EXPONENTIALS. 



THE type of integral which occurs most frequently in the mathematics necessary to the 

 Kinetic Theory is 



\u n e~ M "du (i), 



where n is integral. This can be evaluated in finite terms when n is odd, and can be 

 made to depend on the integral 



.(ii), 



when n is even. In each case the reduction is most quickly performed by successive 

 integrations by parts with respect to 2 . Tables for the evaluation of the integral (ii) will 

 be found in Appendix B. 



When, as is generally the case, the limits of integration are from u=Q to w = oo, the 

 results of integration are expressed by the formulae 



2X' 



The following cases of the general formulae are of such frequent occurrence that it may 

 be useful to give the results separately : 



(" e -Tnu* du J_ /"^ 



Jo 2 V km 



J o 



r i 



i e~ hmut u 6 du=j^^ t 



r 15 / 



Jo e ~* m2tt6rfM = 16V, 



Each integral can be obtained by differentiating that next but one before it with 

 respect to km. In this way the system can be extended indefinitely. 



