54 Perkins — Methods of Using the Galvanometer. 



dC 

 Since r. is the variable "branch -^ is the measure of the sensi- 



tiveness of the arrangement. This differential reduces to 



- E /{ 2r, r t + 2r, r, + r 3 r e + r B r 6 + r, r, + r, r 4 + r, 1 + (r^ r 6 + 



r i ^5 '', + r ' *\ + *", ^4 O / ^3 + (f, ^ ^6 + *\ r * ?\ + ? \ ^5 »*.) / ? '4 !' 



when i\ has been eliminated by the relation r l r\ — r 2 r 3 which 

 is true when the bridge is balanced. 



In this equation r 3 and r 4 are independent variables, and it 

 is now essential to find how they should be adjusted to render 



— 5 a maximum. Assuming r 3 constant and differentiating 



the above coefficient with respect to r 4 and setting it equal to 



dG 

 O we obtain a value of r, which must be substituted in — — 5 



dr 1 



and the result again differentiated with respect to the one 



remaining variable, r 3 . Finally solving for r 3 and making use 



of ?\ r 4 — ?\ r 3 and the value already obtained for r A we have 



r„ r 3 and r 4 in terms of the constants ?\ r h and r 6 . These 



values are 



/(»", + r t ) r h r x _ A /(r y +r 5 )?\r e 



i\ = a/ 1j_1 — *L — 5— 1 r 3 = a/ y —>-^ — 5/. i B r l — s / r r * 



V r x -i- r 5 r r a + r 6 



We are uow in a condition to evaluate the sensitiveness of 

 the bridge method by substituting the values of r 2 r 3 r 4 in the 

 coefficient of sensitiveness, reducing it to 



dC b -_E _ 



d^ 1 r A 1 2r B 4- 2r^ V {?\ + r 5 ) + r x } 



where r 6 has been set equal to O, as in either method con- 

 sidered it will be small compared to the other resistances, and 

 if it is retained the above expression is too unwieldy for ready 

 comparison with that which follows. 



The second case is much simpler. Here the battery, variable 

 resistance and galvanometer are arranged in series. Assuming 

 here also that the battery resistance is negligible, and also 

 using the same E.M.F. we have 



C = — ■ and — — — ■ = b 



r, + r b dr x (r, + r 5 ) 



* Heavyside obtained almost the same results in the fourth paper, Vol. I. 

 Electrical papers, but his value for r 2 is clearly wrong. It should read 



\y x v ic — ±_L instead of W x ^ ?', as can be readily shown bv applying 

 ^ g + x si 9 + » 



the bridge formula to the values of r 3 and r 4 and solving for r 2 . Moreover, 



dC 

 the value of the differential coefficient — . was not given, as his method was 



dr\ 

 essentially different from the one just outlined, so his paper was of little 

 assistance. 



