Equation of Electrical Flow. 433 



tan J3 = l and therefore ~- = 



4tt 2 tan /3 

 and the work 



_E 2 1 h . 3 E 2 ^sin 2 /3 



"R 2cos/3 2ta^S Sm ^° r R 4 ' 



which is the expression we should obtain if we integrate the 

 square of the current multiplied by ~Rdt, seen as follows :— 

 In the general expression (a) obtained above for the pro- 



E 

 duct of the projections make a = E, b= p, and l = 0, y=0 



the expression (a) becomes 



E 2 t 



p—- ^ {tan j3 — sin ft cos /3j , 



or 



4R-t^ {tan/3 - sin/3cos/3} ' 



j^ t x sin 2 /3, as above. 



or 



E 

 4R 



Thus the whole of the work goes to heat the wire, and, 

 further, substituting in the equation for E in terms of V, it 

 may be shown to be entirely derived from the charged con- 

 denser. 



The work may be written, eliminating E, 



2V 2 cos/3 t 2 . so 



or 



Now VY 2 = V 2 sin 2 /3, and thus the work is 



Videos 2 /3, 



or, since cos 2 /3= — 



Vi% , . KR 

 - lr ,and^=- F 



Vi 2 K 



which is the ordinary expression for the energy stored in the 

 condenser ; and this appears from the investigation to be 



