Wheatstone's Bridge for measuring a given resistance. 117 



(a + b)(c + d) _ b c+d 



a+b+c+d ~ 'b + d' 



(a + c)(b + d) _ b + d 



a+b+c+d ~~ C ' c + d' 

 and 



d 

 Substituting these in equation (2), we get 

 (ad— bc)d 



W _W V (b+d)(0 + d) 



~ (bc + ef)(b + d)(c + d) + ce(b + d)* + bf(c + d) 2 ' ' [ } 



Now c, d, e, and / being fixed, and b the variable, we have to 



make u a maximum. As Ed is constant, it may be dismissed. 



As to the numerator (ad— be), it vanishes when at a balance ; 



but of course such a thing as an exact balance is unattainable. 



Let d± A be the real value of the resistance we are measuring, 



be 

 d being the calculated value — , and A a small difference, then 



a(d±A)-bc=±aA. 



Therefore the numerator varies as a or as b, since in the present 

 case a and b vary together. Hence we may write b for (ad— be). 

 Thus 



b 



U ~ (bc + ef)(b + d)(c + d)+ce(b + dy i + bf(c + df ' 



du 

 By differentiation and putting —z=0, we obtain 



(bc + ef)(b + d)(c + d) + ce(b + d) <2 +bf(c + d)* = bc(b + d)(c + d) 



+ b(bc + ef) (c + d) +2bce(b + d) + bf(c + d)*; 

 therefore 



ef[b + d)(c + d)+ce(b + d) 2 =b(bc+ef)(c + d)+2bce(b + d), 

 def(c + d) + ce(b + d) 2 =b' 2 c(c + d)+2bce(b + d), 

 b*c(c + d+e)=de(cd+df+fc), 

 which gives the relation sought, 



-y/l 



d+df+fi 

 c c + d + e - e > • • • ■ W 



