1871.] of Testing a Telegraph Earth. 183 



2 (2 F + A) ( W -f W") ( (W + W"f 

 + 



.*. r 



2 (2 F + A) -f W + W" 

 W + W" 



2 



Secondly let W and W" be both small compared with A 

 and F, but W and W" not necessarily equal to one another, 

 then 



(2 F + A) (W + W) 



2 (2 F + A) 



approximately, 



W + W" . .■ 



.*. r = approximately. 



(Appendix III.) 



Equation (IV) can be obtained directly from equation (III) by 

 substituting Gr for A, and this is precisely what would be antici- 

 pated since the law for a differential galvanometer, when the cur- 

 rents balance one another, must be precisely the same as that for a 

 Wheatstone's Bridge at balance with equal branches ; the two 

 branches of the Bridge corresponding respectively with the coils of 

 the differential galvanometer. 



(Appendix IV.) 



If x, y, z, be the resistances of the three " earths" used, and 

 a and /? the resistances of the two leading'wires then. 



/ where M is a constant depending on the 

 IVt \ 

 a ° = < battery power employed, and the deli- 



' cacy of the galvanometer. 

 M 



b° 



G + F + a + j3 

 M 



G + F + 



x + 



M 



y + 



a + 



G + F + 



x-H 

 M 



z -f- c 



^ + P 



G + F + y + z + a+0. 



23 



