396 



Prof. Thomson on Transient Electric Currents, 



where p and fJ are the roots of the equation 



Using equations (3) and (1) to determine the arbitrary con- 

 stants K and K', and to derive an expression for 7, we obtain a 

 complete solution of the problem which is expressed most con- 

 veniently by one or other of the following sets of formulae, 

 according as p and p' are real or imaginary : — 



9 -■{(... |).".(.A-|).-} 



7= 

 where 



Q .- 



2«AC 



CA/ ' 



(6), 



q=-^ e""^' -Ta'A cos {u't) + | sin {cc't) \ 



7= 



Q -A 



t'AC 



€ 2A . sin {a't) 



where 



VGA 4AV 



(7). 



Among numerous other beautiful applications of his " electro- 

 dynamometer," Weber has shown a method of determining what 

 he calls the '' duration ''* of a transient electric current. In 

 accordance with the terms he uses, the duration, and the mean 

 strength of a transient current may be defined respectively as the 

 duration and the strength that a uniform current must have to 

 produce the same effects on the electro-dynamometer and on an 

 ordinary galvanometer ; so that if T and T denote the duration 

 and the mean strength of a current, of which the actual strength 

 at any instant is 7, we have 



(S); 



fy,t 



f^rydt 

 ^ 



* " Bestimmung der Dauer momentaner Strome mit dem Dynamometer 

 nebst Anwendung auf physiologische Versuche/* § 13 of Weber's Electro- 

 dynamische Maasbestimmungen, Leipsic 1846. 



