in the Branches of a Wheats! 'one's Bridge. 167 



Adding (3) and (4) we have 



Ii!3[R?+lV+k!+lV] 



= l£±[Ti 2 2 + Llcot+nl + LtG>s] (8) 



SubtrMcting (8) from (7) we have 



Ii!.s[RiB 3 + L X L 3 a> 2 ] = I| 4 [R 2 R 4 + L 2 L 4 <*>*] . . (9) 



This may be written so as to express the ratio of the currents 

 in the branches as 



Ii,3 _ R 2 R 4 + L 2 L 4 co 2 



-2,-1 



R1R3 + LiLg CO* 



Equations (3) and (4) may also be so written as to express 

 this ratio, and we have 



Ji!s Rjj + Ll o> 2 _ R 4 + 14 <» 2 _ R 2 R 4 + L 2 L 4 to 2 

 I| 4 ~K+U *> 2 Kl + LI *> 2 ^i R 3 + LxLs a> 2 



j! J 2 



_^_2 «4 



J 2 J 3 



(11) 



In experimenting with a Wheatstone's bridge with reference 

 to its use with the alternating current for the purposes of 

 developing a new range-finder for coast defence, this whole 

 problem of the "Wheatstone's bridge in its more general form 

 for alternating currents has come under our consideration. 

 In looking over the literature of the subject nothing has come 

 to notice which treats so explicitly of this general problem as 

 of the particular case of its use with the direct current. 

 Although Professor J. J. Thomson treats the subject in his 

 recent work, e Experimental Researches in Electricity and 

 Magnetism/ and there derives the conditions for zero current 

 in the galvanometer, yet the graphical method of viewing the 

 problem adds so much to the bare analytical statements, 

 especially where one is experimenting with the bridge and 

 desires to know how any particular quantity varies for a given 

 variation of any other quantity, that this discussion became 

 desirable. In particular, it was desired to know how a varia- 

 tion of the inductance in one arm of the bridge affected the 

 inductance of a second arm, other constants remaining un- 

 changed, while the galvanometer continually indicates zero. 



The above equation (11) contains the solution of this, but 

 it may be transformed so as to represent more clearly the 

 exact relation between the variables. Let us assume that all 

 the constants of the bridge except the inductances L 3 and L^ 

 of branches (3) and (4) (tig. 1) remain unchanged. Let the 



