118 Mr. 0. Heaviside on the best Arrangement of 



and as a — - J) therefore 

 a 



•=v/T 



cd+ df+fc m 



c + d+e ' 6 Kn 



d^u 

 These values of a and b will be found to make -j^ negative ; 



therefore they give the most sensitive arrangement for the fixed 

 values of c, d, e, and /. 



If b vary from nothing upwards, it will be found that u rapidly 

 increases up to its maximum value and then slowly decreases, 

 from which it may be concluded that it is better to use too large 

 values of a and b than too small. 



In case c = d, formulae (6) and (7) become 



«=b=J~°+M. (8) 



V 2c + e 



As a numerical example of these formulse, suppose the resist- 

 ance to be measured ^=1000 ohms, the galvanometer e=500 

 ohms, the battery resistance / = 100 ohms, and we make 

 c=1000 ohms; then the best values for a and b will be found 

 to be \/240,000 = 100 \/M, or nearly 500 ohms. 



Having thus determined the relations of a and b to c, d, e } 

 and/, the latter resistances being fixed, we now proceed to the 

 second part of the problem, to determine the best values of a, 

 b, and c when only d, e, and/ are given. This is the case which 

 occurs so often in practice, when we have a battery, a galvano- 

 meter, and a resistance to be measured, and three sides of a 

 bridge to which we may give any values we choose (within cer- 

 tain limits). 



Insert the values of a and b, as given in equations (6) and (7), 

 in equation (5); then, after some reductions, we obtain u= 



ad— be 



2de(cd+df+fc) +5 (c+d+e) (cd+ df+fc) + cdel a A . cd+df+fc • 



V c c+d+e 



We must now consider c the independent variable, a and b 

 being dependent variables, (ad— be) still varies as a. It does 

 not, however, vary as b, but as the product be or ad, since d is 

 constant. Therefore we may put the known value of be in the 

 numerator instead of (ad— be). Thus u = 



v 



cde ~d + df+fc 

 c + d + e 



2de(cd+ df+fc) + {(c + d+e) (cd+ df+fc) +cde\ x / ci . cd + df+fc 



3 V c c+d+e 



