Certain Problems of Two- Dimensional Physics. 775 

 Thus with the previous value of V» we have 



V =i{^)+/(T)}4-i : {F(^)-F(T)} + Mt(X 1 ,Y 1 ) + ^(X 2 ,Y 2 )} 



2c 



+ 



2c) A 1 BX X BY/^ 



where /and F are functions to be determined by the form of 

 V at infinity, and by the conditions that 3Vi/df =BV t /d?7 = 

 at the singular points of the transformation, where 



Y^X/, Y 2 '=-tX 2 '. 

 But 



=0, 



at the singular points. In the same way the differential 

 coefficient of the same function with regard to r vanishes at 

 these points. And it is obvious that the differential co- 

 efficients of ^(x,y) with regard to f and rj also vanish at 

 the singular points. Hence c)V /Bf and BVq/^t; vanish 

 at these points. 



This result greatly simplifies the work of obtaining the 

 potential of a cylinder of any given form. For example, 

 let p be constant and let 



Then 



KX^ + X^ + Y^ + Y, 2 ) and ~ T (XY'-X'Y)^ 



contain ?? only through multiples of cos ??, cos 2t7, 

 ...,cos (71 + 1)77. And therefore V is of the form 



- 2irp(a?- V-26 2 2 ... -nh?)^ + A^-^cost;+ ... 



+ A, l+1 ^-^ + 1 )?cos(n4-l)7 7 }, 



and both dV /Bf and 'ftVo/dv must vanish when 



