Method of Measuring Battery Resistance. 521 



d ; thus, since A— Q=e—(c-\-d)u, 



B _ I)= (ac-bd)u + eb } 



a + b v y 



So, if ac = bd, the difference of potential B — D is quite inde- 

 pendent of the current through the cell (except in so far as the 



electromotive force e depends upon it) and is equal to 5 



or -7, which are the same thino-. 



c + d' ° 



The current u is of course dependent on the resistance r of 



the branch AC, being 



u= (a + b + r)e . r , 



(a + b)(c + d)+r(a + b + c + dY ' * ' V J 



so we may also write the above difference of potential in terms 

 of this resistance r, thus : — 



B-D = e (a + 6>4-r(6 + c) Ub) 



(a + b)(c + d)+r(a + b+c + d)' ' v J 



All the differential coefficients of this with respect to r 



contain the factor ac — bd; consequently when this factor 



vanishes this quantity is independent of r. 



Conditions for Sensitiveness. 

 To find out what are the values of a, b, and c which give 

 the greatest sensitiveness, we can subtract the value of B — D 

 when r is infinite from its value when r is zero, and can make 

 this quantity a maximum when the condition ac = bd is nearly 

 fulfilled. The quantity which has to be a maximum is 



y= (B- D)B -(B-D). = (< + jg-ff e + <0 . . . (6) 



The resistance d is supposed to be given ; so let us define the 

 others with reference to it, putting 



c = \d, a=pd, and &— \fjid = \fi{l—z)d, 

 where z is a small quantity ; then the above quantity becomes 



.... (6a) 



'_ (X+lJV + 1) 

 Considered as a function of X, this is a maximum when \ = 1 ; 



it has in fact the same value whether X=n or -. Considered 



n 



as a function of fi, it has no maximum, but it is greatest when 



fi is infinite, though it does not increase fast after p is tolerably 



large ; the curve is, in fact, a rectangular hyperbola with 



asymptotes y=l and fi=— 1; and 1 is its greatest value for 



positive values of jjl. Accordingly the most sensitive arrange- 



