282 Mr. J. H. Jeans on the Vibrations 



B, so long as the relative velocity is comparable with the 

 average in normal air. 



§ 6. A more convincing proof of the smallness of the in- 

 tegral in (8) is supplied by Cauchy's theorem. Imagine the 

 function Y evaluated for all values of t real and complex, 

 then we have 



J V^^^'^f = 2/7rSR, (12) 



Avhere the inteo-ral on the left-hand is taken from ^= — cc to 



o 



f = -f CO along the real axis of t, and then back from ^= -f oo 

 to t=- —GO along a semicircle of infinite radius, having f = 

 for centre. On the right-hand ^K is the sum of the residues 

 of the function \e'P* inside this semicircle. 



The first part of the integral is X+/Y. The second may 

 be written 



and the integrand, in general, vanishes through the occurrence 

 of the factor e~'P^^'^^, except near 6 = (), 9 = 7r, and here the 

 contributions to the integral vanish on account of the factor 

 VdO. Lastlv, if the function Y has an infinity occurring 

 at t = a + i fi,\\it\i residue u, XR may be w^ritten ^e'P^'^+'-^hi, 

 the summation extending over all infinities for which /3 is 

 positive. Equation (12) accordingly becomes 



X + zY = 2i7rS£^~^/3(cosj^a: + zsinj9it)?^ . . (13) 



By hypothesis p^ is large, and it is also positive. Tbus 

 the smallness of X-f2Y is guaranteed by the occurrence of 

 the factor e~p^. 



§ 7. To take a definite example,, suppose 



1 

 Y = 



a--^t- 



X-f iY=— I ~^Jt = -^e-"P, 

 a/)J_^ a--\-t- aap 



and the energy of vibration ^ajo-(X" + Y')^ is 



TT- 



e-'-'^P (14) 



The appropriate unit of time in this case is of course a. 

 If p in these units has a value 200 we see from (14) that 

 the " elasticitv '^ of the molecules has introduced a factor 



z>-400 



