484 Prof. 0. W. Richardson on the Theory of 



Hence by comparing coefficients, <3>(0) = and 



f (0) = I*Y0), *'(<>) = j^ *"(0), . . , 



r~K0) = ^*"(0), & (25) 



Since 



^O*)=^(0) + ^'(0) + ^+"(0) + ... . (26) 



the relations (25) are sufficient to determine ty(v — v ) i£ it 

 exists. 



Let us now consider the equation which expresses the 

 balance of kinetic energy between the outgoing and 

 returning streams of atoms. This may be written, using 

 (2), (6), and (16), since E = 2NRT, 



J = v -h* ^°' v) Mv= - NRT= 2RT3>(T>~ m \ . (27) 



-0 ^ET_ X 



or 



J 00 hv kvc, 



X(v-v )e~ rt 7^ = 2RT$(T>~kt , . (28) 

 "0 

 if x( „_ Vo )=T„^(v-v ) .... (29) 



is a function only of (v— j/ ). This assumption is suggested 

 since the form of (28) is the same as that of (20). Thus, by 

 the same treatment as in dealing with (20), 



:(0) + x^°)+(x)V (0)+ • • • +(ir)V(°)+ • • • = 2 *( T ) 



= 2RT{f(0) + (^)t'(0) + (x)V'(0)+ • • • +(x)V"(0) + - ■ • } 

 whence ^(0) =0 and 



x'(0)/+(0)= x "(0)/^'(0)= .... = x »(0)/^«-i(0)= . . . = 27*. 



. . . (30) 



Since x(#)=x(0) +* X '(0) + £ % "(0) + . . . 



97. 



-3 



= 2/{,^(0) + |!^'(0) + f,f "(0) + . . .) 

 = 2AJ" ^-(.«)^=T a .^( i K), from (29), 



