1886.] Circular Discs by means of BesseVs Functions. 431 



Also 

 i I fie~^ s sin p,vJ [ftp) dp, = imaginary part of —5 



{{z-cvf + p^ 



= ^1 ( 2 sin 3 % - V cos 3%) 



f 



JO 



'* z \u cp (u) sin \v sin pvJ (Xu) J (pp) dp (14), 



then V = (p (p) when z = and p > c \ 



dV \ (15). 



-j- = when z = and p < c \ 



8. In the last article we have quoted the known formula 



2 f 00 

 J (\) = - sin (A, cosh 6) dd. 



7T Jo 



This result may be easily established by comparing the results 

 obtained by integrating the definite integral 



* 00 /> 00 



I du I cos# cosm 2 (« 2 — \ 2 )dx 



Jo Jo 



with regard to # and 1* respectively. If we first perform the 

 integration with respect to u, the result 



1 /it f™ cosxdx 1 fir f A cos#c&c ,_ rtN 



t+oa/o — ri (16). 



2 V 2J A (^_x 2 )^ 2V 2J (x, 2 -^ 



The second integral on the right-hand side is equal to 7rJ (X.)/2; 

 and if in the first integral we put x = \ cosh</>, and afterwards 

 make e* successively equal to 2a.ia 2 and l/2\u 2 and add the results, 

 we shall find that 



[°° cosocdx f 00 / „ „ , 1 \du /im 

 i= cos XV + j-, — (17). 



