Mr. R. S. Brougb on Wheatstone's Bridge. 23 



the galvanometer) is a merely mathematical problem destitute 

 of physical meaning. Clearly the proper procedure is first to 

 prove that there exists a general maximum when balance is not 

 established, dependent on the resistances of the four branches 

 and on the resistance of the battery, and then to find to what 

 condition this converges as we approach balance and becomes 

 equal when we reach balance. 



From the second of the above equations we find that the ge- 

 neral maximum is given by 



E= (« + 6)(g + «*) + (ad- be) 2 



a+b+c+d {a + b + c + d){(a + c){b + d) + {a + b + c + d)f}'' 



which, as balance is approached, rapidly converges towards the 

 condition 



E= {a + b)(c + d) _ 

 a+b+c+d 



This is the general useful solution, since it is only near ba- 

 lance that it is of practical importance that the condition of 

 maximum sensibility should be fulfilled ; and we notice that it is 

 independent of the battery resistance. 



The resistance of the battery has been supposed to be of finite 

 magnitude compared with the other resistances in the bridge. 

 If, however, the resistance of the battery be indefinitely large 

 compared with the sum of the four resistances in the bridge, 

 then the condition 



{a + b){c + d) 



E = 



a-i-b + c + d 



holds universally, i. e. is independent of whether balance is nearly 

 established or not. (Such a case could, of course, only occur 

 in the physical laboratory.) 



Again, if the battery resistance be indefinitely small compared 

 with the sum of the four resistances in the bridge (as in measu- 

 ring the " insulation " of a well-insulated telegraph-wire), we see 

 that the condition 



™ ac Id 



^ = — I ^TTH 



a+c o+d 



is universally true, i. e. is independent of whether balance is 

 nearly established or not. 



Of course when balance is nearly established the two condi- 

 tions become identical. 



Bridges for testing telegraph-lines are almost invariably 

 wrongly arranged. It can be proved* that when the galvano- 



* See Clerk Maxwell's ' Electricity and Magnetism/ Art. 348. 



