Theory of Galvanic Polarization. 433 



theory of surface-energy which Gauss has made the founda- 

 tion of the doctrine of capillary action, in a case in which all 

 the circumstances of the forces are known, and as it also 

 illustrates some other points, the direct calculation is here 

 appended. 



Consider, first, an infinite plane electrified surface, and 

 imagine a straight line drawn across it. The mutual repulsion 

 of the electrified parts on the two sides of this line will result 

 in a tension tending to tear the parts of the surface asunder 

 along the line, and whose intensity, measured across unit length, 

 we can calculate as follows. 



Imagine a unit of electricity situated at a point distant £ 

 from the line of division ; the repulsion exerted on it by the 

 other half of the electrification is easily expressed in polar 

 coordinates, r, 6 ; 6 being measured from the shortest distance 

 to the line of separation. 



It will, however, be more convenient to take this unit charge 

 at a distance k from the plane, and to measure r, 6 from its 

 projection on the plane as pole. The repulsion exerted on it, 

 resolved parallel to the plane, is 



V- • pcose 



i ir \" 



2 _^_ '2 y 



(h 2 + r 2 )i /n 



where p is the surface-density of the electrification, and 

 cos <£ = |; jr. 



Therefore the repulsion 



J* (r 2 + h 2 )i 



To integrate this, write 



r 2 -% 2 = (r 2 + h 2 )z 2 ; 



therefore 



r 2 + h 2 



rdr= -= » zdz; 



1 — r 



and the integral 



*2 



dz 



1- 



-r 



This quantity becomes infinite at the upper limit ; so that 

 for an infinite plane sheet the tearing-force due to the electri- 

 fication would be infinite ; a result which w r ould also follow 

 readily from simple consideration of the dimensions of the 

 variable involved in the integral. 



Suppose, how T ever, we take a finite sheet bounded on the 



