239] Rate of Dissipation of Energy 201 



semicircle BPA, over which R is constant and infinite, while 6 varies from 

 to TT. On this part of the path we have, since t' = Re ie , 



dt' = Reside, 



hence this part of the integral becomes 



t 



I = JJ'gipRcosO-pRsmd Re^idf) 



J 



0=0 

 and the original equation (485) assumes the form 



(486). 



To evaluate /, we divide it into three parts I lt / 2 and 7 3 , the first being 

 the integral from 6 = to 6 = e, the second from = eto0 = 7r e, and the 

 last from 6 = TT e to # = TT. The integrand can be written in the form 



U' {i cos (6 + pR cos 0) - sin(0 + _p.Rsin 0)} e -P-Esine+io gJ B (437). 



Let us choose the quantity e so that e shall be vanishingly small, whilst 

 -Rsine shall be infinitely great in comparison with logjR, this being always 

 possible when R is infinite {e.g., we may take e = R~* so that R sin e = R%\ 

 Giving e such a value, it is clear that e~ pSsin8+l KS will vanish from = e 

 to 0=Tr e and therefore, except in the special case in which U' becomes 

 infinite within the range in question, we shall have 7 2 = 0. 



The cases in which U' becomes infinite within the range of integration 

 must now be examined. It is clear that if, when R = oo , U becomes infinite 

 less rapidly than an exponential e* R , then by taking R sufficiently great, 

 the integrand (487) will always vanish. When U becomes infinite with 

 the same rapidity as an exponential this is not in general true, and 

 the same is the case if U becomes infinite more rapidly than an ex- 

 ponential. The important case in which the result is not true is that in 

 which U contains a term of the form cos (qt + e), which gives rises 

 upon the semicircle to terms containing the factors e gR sin 9 . If q < p, the 

 result 7 2 = will be seen to be true ; if q > p the result 7 2 = is not true, 

 but the main proposition can be proved by a slight modification of the 

 present proof; if q=p the main proposition is obviously not true. On 

 general principles it will be seen that this is the only important case of 

 exception. Physically, this possibility represents vibrations " forced " in one 

 molecule, by vibrations of equal period in the second molecule. These 

 vibrations, then, form an exception to the proposition which we are trying to 

 prove, that the vibrations excited by collision are small in comparison with 

 the energy of the exciting agency; but the physical deduction we wish to 

 draw from this proposition will be in no way invalidated, for vibrations of the 

 kind in question do not represent a transfer oi energy between translational 



