186 ELECTRICAL MEASUREMENTS 



For suppose the actual value of P differs from that necessary 

 for a perfect balance by an amount 5P, then referring to equations 

 (1) and (2) and substituting the approximate values of I B , 

 __ -EMdP _ 



IG > == ^,, f . ywy.px , (M+N)(X+P)(N+P)(M + X) 



R G (M+N)(X+P) + - M + N + X + P~ 



-EMdP 

 ~ Denom. i 



and 



EM5P 



EMdP 

 Denom.2 



The better arrangement will be the one corresponding to 

 the equation having the smaller denominator. 



Denom.! - Denom. 2 = Ro[(M + N)(X + P) - (M + X)(N+P)] 

 = Ro((M - P)(X - N)] (7) 



From this it is seen that if the opposite resistances are equal, 

 the sensitiveness is the same for both arrangements. If the 

 algebraic sign of the bracket is -f- , Denom. 2 is less than Denom. i 

 and the second connection should be used; if it be , the first 

 is to be employed 



The better arrangement is the one where the galvanometer 

 joins the junction of the two highest resistance bridge arms to 

 the junction of the two lowest. An exact discussion of the 

 more complete formula which includes the e.m.f. and the resist- 

 ance of the battery gives 



Denom.! - Denom. 2 = (R G - B)(M - P)(X - N). (8) 



Considering the battery and the galvanometer, the one having 

 the higher resistance should join the junction of the two highest 

 to the junction of the two lowest bridge arms. The importance 

 of the proper relative position of the battery and galvanometer 

 increases with the disparity of the bridge arms. 



As an illustration, consider the bridge shown in Fig. 106 where 

 there is a great difference in the bridge arms. Denom. i - 

 Denom.g is + , so the second connection should be used. 



