240 Sir J. J. Thomson on the 



corpuscle from the system with which it collides, 

 d 2 x _ At 



1 !dx*\ 1 fi J 1 ., 



where v is the velocity of the corpuscle before the collision. 

 If we take t = when the corpuscle is closest to the molecule, 

 we get by integrating this equation 



\ mxr) 



f(t) the acceleration of the corpuscle is equal to p/mx 8 , and 

 thus 



\ mv J 

 And /j, C+™ cos 7< t it 



\ mt; / 

 To calculate the integral, let 



cos ^ dfe 



(FT?)!' 



By differentiation, and integration by parts, we easily find 

 1 du d 2 u 



-r 



</ £^ dq 2 C * U ' 



ov it cq-=x, 



1 <fw rf 2 ?< 

 A' <£& dx 2 

 If u = xw, this equation becomes 



d 2 u? , 1 dw /- 1\ rt 



^+7^-H 1+ .?) =0 - • • • (1) 



Now the solution of the equation 



<Py , 1 dy A n 2 \ . 



is (since this is Bessel's equation with u? for the independent 

 variable) 



AI„(.r) + BK„(.r), 



