157] ELECTRICAL IMAGES. 305 



The function V is everywhere continuous, for W and W are 

 continuous except at S and there, by (3), 



(10) Vi=Wt+ W{ = W e '+ W e =V e , 



so that V is continuous. 



We have already seen that the derivative of W is continuous 

 in crossing 2, and accordingly that of W is continuous in crossing 

 2'. Now the derivative of W is continuous in crossing 2', since 

 W is defined by the same continuous analytic expression in T' and 

 T", and the derivative of W is continuous in crossing 2, since W 

 is defined by the same continuous analytic expression in T and 

 T". Accordingly the derivatives of V as well as V itself satisfy 

 the required conditions of continuity, and on S, since 



x x', y = y, z = z f , I = R, X' = x, 



we have F=TT and F, the function assumed, is therefore the 

 potential of an equilibrium distribution on the bowl. At the 

 center of the bowl we have on the one hand 



while in order to employ the formula (9) we have x = y z ^, 

 X = c 2 . 



But when I is infinitely small, X' must be infinite of the second 

 order, as we see by making I infinitesimal in the equation 



<*> 



The terms of the lowest order are 



V/4 



^+^+^=^. 



Hence approximately 



x'-* 4 



I* ' 

 and 



TF'(0) = limf (f -tan-'f ) = Ihn tan-| = |. 

 I=Q i \4 iaj I = Q l iff A 



Therefore we have finally 



? + tan->%| = -l, 



(13) e = 



W. E. 20 



