432 Mr Basset, On the Potentials of Circular Discs. [May 10, 



In the same way we can show that 



C sin cc o oo C 



I r= I sin (A, cosh <b) deb 



A (x?-\*Y Jo 



(x 2 - \J 



}\di 



= f mnp*+ Mu 



.(18). 



Whence, the right-hand side of (16), 



2V 2Jo 



cos W + 1—5 V 



1 \ du t 7T§J n (X) 



4V2 



4w V it 



.(19). 



Now if we integrate the original definite integral, with respect 

 to x, the result 



- 1 y 1 r c ° s ( xv + w) t + § y 1 /.* sin ( xv + ») * • 



whence by (18) and (19), 



2 



7T Jo 



sin(\ cosh cb) deb. 



9. As an example of the formulas of Art. (4), consider the 

 case of an uninfluenced disc which is electrified to unit potential : 

 then 



O * CO 1* co pCC pOO 



V=-\ d\\ du\ dv I e~^ s u\ cosXv cos fivJ Q (\u) J (fiv)d/u,. 

 tt J Jo Jo Jo 



r 3 



Now ue- pu J (\u)du = 



Jo 



Qf+PY 



Therefore 



/.OO /.CO 



I cfo* I ttX cos\vJ (\u) dX = lim of 

 Jo Jo 



Put A = /Sy, and the integral 



\{3 cosXvdX 



\jo (p+xy j^=o 



r 



cos ftyvdy 



Hence 



= lim of 



(l+3f) , J/»=o 



= 1, when /3 = 0. 



2 f 00 e - ^ 

 F= - I sin ixcJA^p) da, 



7TJo ^ 



