t534 Mr. Jackson on the Method of Transmission of 



This expression is to be integrated over the infinite cylinder, 

 and then it is clear that the deposit on any portion of the sides 

 is proportional to its area. The area of unit length of the 

 cylinder is ird, where d is the diameter, and the value o£ BY 

 expressed in terms of r, 6, (j> is r 2 cos <fi 86 8$ 8r. Denote the 

 number of particles reaching unit length o£ the sides by N, 

 we have 



_E 



The limits of integration being 



#, —cos "M — cos 6 ) to cos _1 1 - cos cj)J 

 r, to dsec<j), 



7T 7T 



Performing the first integration with respect to 6, we have 



TV 



r- 7> f d see <J5 R / „.2 ni 



N^Mrf " e '* cos 2 d>[l- T2 cos 2 <l>Ydrdcl>. . (2) 



Jo Jo 



In this integral substitute r = xd sec (/>, whence 



C 2 C^- — T x d sec , , 



^=Md 2 | I e L cos 2 (/>(l-.r-;i-^rf</). . (3) 



«- o Jo 



The maximum value of this integral is obtained by omitting 

 the exponential factor, and corresponds to a vacuum when 

 all the activity would reach the sides. Denoting the maximum 

 value of N by N , we find, as is otherwise obvious, that 



N =i7rMd 2 (4) 



Let the ratio N • N be denoted by s. 



An effective approximation to the integral expression for the 

 value of 5 may be found by expanding (1— x 2 )* in powers of x. 

 Introduce the notation 



Jn(z)=\ 1 e_ zx x n dx, . ... (5) 



and write 



h=t<1 (6) 



