Resistance of Bismuth to Alternating Currents. 305 



This current is represented by OM in fig. 4. 



If now a small resistance r be added to the resistance R 



r 

 in AC, a second current having the value Kyrsin0 — repre- 

 sented in fig. 4 by OL — will pass through the galvanometer- 

 arm. These two currents will have a resultant the magnitude 

 and phase of which is given by OT. 



This resultant current being alternating will produce no 

 deflexion of the galvanometer; but let a two-part commu- 

 tator, attached to the dynamo-shaft making one revolution 

 per alternation, be introduced into the galvanometer-arm, 

 and so arranged that the brushes can be set to commutate at 

 any desired phase, <£, of the current supplied to the bridge — 

 (p is represented by the angle LON in fig. 4. In general 



the galvanometer will now show a deflexion, but there will 

 be one position of the brushes which will completely com- 

 mutate out the galvanometer current, and reduce the de- 

 flexion to zero. This will clearly happen when commutation 

 takes place a quarter of a period ahead of or behind the 

 galvanometer current, that is when the line ON in the diagram 

 is perpendicular to OT. When this is the case it is at once 

 seen from the diagram by projecting MT on OL that 



r e 



Y = ^ (cos a + sin a tan (/>). 



It having been found by experiment that for a given fre- 

 quency e is proportional to E, this equation shows that if the 



e r 



frequency — and therefore ^ and a — be fixed, t* is a linear 



function of tan^>; so that if a series of corresponding values 



