288 Mr. T. Gray on arranging Wheat stone' s Bridge 



Substituting this value in (13), we obtain 



(E+/)(/R-a 2 ) 2 = (a-/)^E(R/+a 2 ); 



.-. (R 3 + 2R 2 «-3Ra 2 y-tf 2 (3R 2 -2Ra-a 3 ) = 0; 



,\ (R-a){R + (R + 3a)-a 2 (a + 3R)} = 0. 



And R=a is one root. There remains the cubic equation 



a* + 3a 2 B,-3aRf--Wf=0, .... (15) 



which, when solved in the ordinary way, assumes an irredu- 

 cible form. It is more convenient to retain the equation as it 

 stands, and solve by approximation for the numerical root in 

 particular cases. 



It will be noticed that R = a is not a root which generally 

 satisfies the equation, because, by equations (12) and (14), 

 /= R in that case, which is contrary to the hypothesis of / 

 being a constant. Equation (15), however, provides a maxi- 

 mum value of GV g ; and hence this is the root required. 



The following table shows the calculated values of a, c, and g, 

 corresponding to the battery-resistances (/) and the resistances 

 to be measured (R), given in the first and second columns re- 

 spectively. The advantage obtained by increasing / is plain 



from the increased value of 



i.e ' 



as shown in the last column. 



The same thing may be readily observed by examining the 

 fifth set of curves. The abscissas of these curves represent 



battery-resistance, and the ordinates the value of 



GV# 



i.e 



I 



E. 



a. 



Co 



9- 



i.e 



1 



0-01 



0-42 



0-065 



0176 



0-0123 



„ 



010 



044 



0-20 



0-194 



0-0417 



,, 



1-00 



1-00 



1-00 



1-000 



0-0625 



„ 



10-00 



2-30 



3-73 



5-290 



0-0424 



,, 



10000 



6-23 



11-70 



38-810 



0-0140 



» 



1000-00 



18-76 



36-12 



352-000 



0-0052 



10 



o-io 



1-60 



0-188 



0-256 



0-054 



,, 



1-00 



4-38 



1-630 



1-918 



0-131 



;> 



10-00 



1000 



10-000 



10-000 



0-197 



> j 



100-00 



22-70 



37-06 



51-53 



0-142 



» 



1000-00 



62-00 



121-00 



384-40 



0-054 



50 



10-00 



28-6 



14-8 



16-36 



0-343 



>> 



100-00 



630 



77-0 



80-00 



0-427 



» 



1000-00 



1510 



2620 



459-00 



0-469 



