Approximate Spheres and Cylinders. 179 



Tbe electric charge Q, reckoned per unit length of the 

 cylinder, is readily found from (2). We have, integrating 

 round an enveloping cylinder of radius r, 





g«tf=-S>, (6) 



and Q/(/>! is the capacity. 



We now introduce the value of Su/u from (5) into (4) 

 and make successive approximations. The value of H % is 

 found by multiplication of (4) by F„, where w=l, 2, 3, &c, 

 and integration with respect to 6 between and 2-n-, when 

 products such as F^, F 2 F 3 , &c, disappear. For the first 

 step, where 2 is neglected, we have 



O = H^ o "JF n W-^H aJF n W,. .... (7) 

 or 



H*V=H o a (8) 



Direct integration of (4) gives also 



+3H 3 «o'F 3+ ...} +i H o jg0, ... (9) 



cubes of C being neglected at this stage. On introduction 

 of the value of H n from (8) and of Bu from (5), 



^ 1 = -H log(^)+iHo{3C 1 2 + 5C 3 2 + 7C 3 2 + . . .} . (10) 



Thus 



</> 1 /Q=21og( Wo 6)-i{3C 1 2 H-5C 2 2 + 7C 3 2 +...} . . (11) 



In the application to an electrified liquid considered in my 

 former paper, it must be remembered that u is not constant 

 during the deformation. If the liquid is incompressible, it 

 is the volume, or in the present case the sectional area (a-), 

 which remains constant. Now 



= ^{l+f(C 1 2 + C 2 2 + C/+...)} ) 



u 



so that if a denote the radius of the circle whose area 



« 2 = a- 2 {l + f(C7 + (.Y + cy+...)}. . . . (12) 



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