176, 177] STEADY FLOW IN CONDUCTOKS. 349 



The work done in increasing the radius of the disk by dp is, since 

 the mass is increased by %7rcrpdp, 



so that the whole energy of the distribution is 



8 8m 2 



W = 2-TTo- ( a V p pdp = 87T<7 2 { a p*dp = 

 Jo J o 



Inserting this in the value of R 2 



--. 



J n O TTd 



_ 8 



we get 



- 



Q 



STT 



Consequently the infinite conducting mass necessitates a cor- 

 rection equivalent in value to an increase in the length of the 

 wire of between Tr/4 and 8/3-Tr, that is '785 and '849, times the 

 radius of the wire. Lord Rayleigh has succeeded in bringing the 

 limits still nearer together, and the results have been confirmed by 

 experiment. 



177. Current Sheets. The current-density being a solenoidal 

 vector, all that has been said about lines and tubes of such vectors 

 may be applied to current lines and tubes. The current tubes 

 may be defined by the intersection of two families of surfaces. A 

 current sheet will be defined as a portion of space bounded by two 

 infinitely near parallel surfaces, in which currents flow, converg- 

 ing to or diverging from certain points called electrodes. If the 

 equation of the surface is q s = const, and q l and q z are two coordi- 

 nates forming an orthogonal system, the flow may be defined by 

 either the potential V or the current-function "SP, which both 

 satisfy the equation, 104 (7), 



dq l (Aidg dqtfa dq 2 ) 



Problems of plane current sheets may be at once solved by the 

 method of functions of a complex variable, and from them any 

 number of problems for other surfaces may be solved by finding 

 conformal transformations. Such transformations may be found 

 practically for a surface by constructing it of thin metal, intro- 



