Purser — Applications of BcsseVs Functions to Physics. 45 



We have now to determine the constant y. For this purpose, 



consider the value of in the sheath at upper plane close to 

 dr 



circumference of circle, radius R. Here evidently we have 



= 0. 



dr 



Similarly, close to circumference of circle, radius R\ we have 



dr 



Hence, q. p., 



f^ = o 



dr • 



at point midway. 



This gives for r = ^ (i2 + R'), 



n {A,Jlnr) + B,,K,{nr)) + + = 0. (53) 



Now, for this value of r, 



Y,nr c= e Y,{nR), 



Klnr)^e'''^K,{nR'\ q.p. 

 A,,Y,{nr) + B,,K,{nr) = e {^^^P -f 4y£/nHQ^ , (54) 



where 



P = nRK,{nR)Yl7iR) - 7iR'S',{nR')Y,{nR'), 

 Q = nRElnR)YlnR) + nR'K,{nR')Y,{n'R). 

 Remembering that 



K,{nR)Y,{nR) - K,{nR)YlnR) = ^, 



we find P=-l, Q = -l, q. p. 



Our equation to determine y then becomes 



j+yl- 4yR^ 14- - 8y2 = 0. (56) 



In this expression, the term 



nB 



C-\nB O-yJi 1 + 

 \ - e 



