116 Mr. 0. Heaviside on the best Arrangement of 



presented by «=Ex 



(a + b + c+d)(ad— be) 



\(a + b)(c + d) + (a+b + c + d)e\ \(a+e)(b+d) + (a + b + c + d)f\ 



in which E is the electromotive force of the battery, (ad— be) 

 may be positive^ negative, or nothing, in which last case u = 0, 

 and a balance is obtained, no current passing through the gal- 

 vanometer. 



Dividing both numerator and denominator of (1) by 



(a + b+c + d)*, 

 it becomes 



ad— be 



W = Ex a+b+e+d (2) 



J (a + b)(c + d) c \f (a + e)(b + d) \ V ; 



la+b+c+d ~*~ f\a + b + c + d~*' J J 



from the form of which it may easily be seen that, the best value 

 of the resistance of the galvanometer e, when a balance is ob- 

 tained and the other resistances are fixed, is, as Schwendler has 

 shown in the Philosophical Magazine for May 1866, 



(a + b)(c + d) c + d 



e - ( a + b + c + d)~ b + d' ' ' ' " {0) 



that is, the resistance of the galvanometer should equal the re- 

 sistance external to the galvanometer, being the joint resistance 

 of the two parallel branches (a + b) and (c + d). Also it may be 

 proved that the best arrangement of the battery is obtained when 

 its resistance equals the external resistance, that is, 



s_ (a + c)(b + d) _^ b + d 



J- a + b + c + d - C 'c + d' ' ' * * K } 



the joint resistance of the two parallel branches (a + c) and (b + d). 



(In passing, I may notice that Schwendler, in the paper above 

 referred to, and also in a later one in the Philosophical Magazine 

 for January 1867, has assumed it to be necessary for the battery 

 resistance to be very small, in order that the relation exhibited 

 in equation (3) may be satisfied. This appears to me to be 

 totally unnecessary ; for the resistance external to the galvano- 

 meter when a balance is obtained is quite independent of/, the 

 battery resistance. In fact the proper resistance for the battery 

 when it is to be most advantageously used is given by equa- 

 tion (4).) 



As in the present paper we are only concerned with such values 

 of a, b } c, and d as produce a balance, or nearly so, one of these 

 four resistances may be eliminated at once. Let it be a. Then 



