84 PROFESSOR TAIT ON THE 



tions made above. For if the rapidity of equalisation of average energy in two 

 systems is of this extreme order of magnitude, we are entitled to suppose that 

 the restoration of the special state in any one system is a phenomenon taking 

 place at a rate of at least the same if not a higher order of magnitude. 



Clerk-Maxwell's result as regards the present question is that, at every 

 typical impact between a P and a Q, the difference of their energies is reduced 

 in the ratio 



VP+Q/ ' 

 so that, if the masses were equal, the equalisation would be instantaneous. 



VI. On some Definite Integrals. 

 25. It is clear that expressions of the forms 



i~ hx2 x r dx / s-Wy'dy and / g ~ hx2 x r dx / e-Wtfdy , 



J J o J X 



where r and s are essentially positive integers, may lawfully be differentiated 

 under the integral sign with regard to h or to k. In fact they, and their differ- 

 ential coefficients, which are of the same form, are all essentially finite. 



As, in what immediately follows, we shall require to treat of the first 

 of these forms only when r is odd and s even, and of the second only when 

 /• is even and s odd, it follows that their values can all be obtained by 

 differentiation from one or other of the integrals 



fz~ hx2 xdx n-**dy = — ^L= 

 Jo Jo J 4hjh+k 



■Jh + Jc 



and 



fir 



fl-^dx fe-Wydy = — d" 



Jh+k 



These values may be obtained at once by noticing that the second form is 

 integrable directly ; while, by merely inverting the order of integration, it 

 becomes the first with h and k interchanged. 



26. In §§ 21, 22 we had to deal with a number of integrals, all of one form, 

 of which we take as a simple example 



^0 



Is/3=/%J- 



