for the Measurement of a particular Resistance. 289 



Part III. 



The third case is that in which all the resistances, except 

 that to be measured, may be varied. 



If we consider the battery only to vary, we obtain the best 

 arrangement when the greatest total current flows through the 

 bridge ; and this leads at once to the equation 



/= b Sh < 16 > 



Now, equations (12) and (13) taken together give the best 

 arrangement when the battery and galvanometer are constants, 

 while (14) gives the best arrangement when the galvanometer 

 is the only variable. By treating these four equations as 

 simultaneous we evidently obtain the best arrangement when 

 every thing may be varied. 



From (14) and (16) we get gf—a'R ; hence from (12) we 

 have a = E; and consequently by (14) g = a. 



Substituting in (13), we find c = R ; and we are led to con- 

 clude that, for the best effect, 



a = b = c = B J = c/=f. 



When the resistance of the electrodes in the battery circuit 

 is taken into account, the equations become (r being the 

 resistance of the electrodes) 



*=^Kf+r), (17) 



e= VB(/+y)(R+gr) (m 



B+f+r ' ' ' V ' 



'«S* <"> 



/= c S + ' < 20 > 



From (17) and (19) we get 



j, « + 



J c + 



Again, from (17) and (18) we get 



G 



Combining (21) and (22), 





c?-<? 



f= n jZTRi- r (82) 



« = R ^ or a = -rn . . . • (23) 



c—H 2H— c v 7 



