Illustration of the Theory of Gases, 443 - 



only one power of q need be retained, it will conduce to 

 brevity to omit the denominator at once, and take simply 



u=u'+2g(v-u') (56) 



Thus if p y <f> and r, 6 be the amplitude and phase before and 

 after collision respectively, 



rcos0=pcos(£ + 2g(v— pcosc/)).^! .- . 



r sin u = p sm <p ; J 



so that 



r 2 = p 2 + 4zqp cos <£ (v — pCOS(f>) + 4zq 2 (v—p COS</>) 2 . 



From this we require the approximate value of p in terms of 

 r and </>. The term in q 2 cannot be altogether neglected, but 

 it need only be retained when multiplied by v 2 . The result is 



p = r — 8r, 



where 



2q 2 v 2 

 8r=2q(vcos(f)—rcos 2 (j))+— — sin 2 <£. . (58) 



This equation determines for a given <£ the value of p which 

 the blow converts into r. Values of p nearer to r will be pro- 

 jected across that value. The chance of a collision at p, <f> is 

 proportional to (y— pcoscf)). Thus if a number of vibrators 

 in state p, (f> be F(/o) dp d<j)*, the condition for the stationary 

 state is 



d<f>\ (v-pcosd)¥( P )dp = 0, . . . (59) 



Jo Jp 



the integral on the left expressing the whole n amber (esti- 

 mated algebraically) of amplitudes which in a small interval 

 of time pass outwards through the value r. 

 By expansion of F(/>) in the series 



FGi)=F(r)+F(r)G»-r) + .,., 



we find 



r ¥(p) dp = F(r) Sr-i^Xr) (8r) 2 + cubes of q, 



pY{p) dp = rF(r)Sr-i(8r) 2 {¥(r) + rF(r)} + cubes of q. 

 * We here assume that the bombardment is feeble. 



I 



