164 Ml'. Louis Sell wen dler on Bifferential Galvanometers, 



The magnetic moment of the coil g, when a current G passes 

 through it, may be designated by Y; and the magnetic moment 

 of the coil g\ when a current G' passes through it, may be 

 called Y'. Both these magnetic momenta are taken with respect 

 to the same needle, or system of needles ; and we may suppose 

 that neither Y nor Y' alters perceptibly when the needle, or system 

 of needles, slightly alters its position towards the coils, which 

 are supposed to be fixed. (This condition will be fulfilled as 

 closely as possible near balance, when the needle is approximately 

 always in the same position with respect to the coils ; and it is 

 only for such a case that the following investigation is of any 

 practical interest.) 



According to the principle of the differential galvanometer, we 

 have 



a^cr.Y-V, 



where a represents the deflection of the needle before balance is 

 arrived at, and which may be positive, zero, or negative, depend- 

 ing on the relative strength of the currents which at the time 

 are acting through the coils, on the relative position of the needle 

 towards the coils, and on the shape and size of the latter. 

 Approximately we have further, 



Y=mUG, 



Y' = m'U'G', 



U and U' being the number of convolutions in the coils g and^' 

 respectively, and m, nl representing the magnetic momenta of 

 an average convolution (one of mean size and mean distance 

 from the needle) in the coils g and g^ respectively, when a cur- 

 rent of unit strength passes through them. 



Further, as the space of each coil to be filled with wire of 

 constant conductivity is given, we have 



U z=n\/g, 



as can be easily proved. 



n and n! are quantities independent of^ and ^' so long as it 

 may be allowed to neglect the thickness'of the insulating cover- 

 ing of the wire against its diameter, which for brevity^s sake we 

 will suppose to be the case. With this reservation n and ?/ de- 

 pend entirely on the size of the coils and on the manner of 

 coiling. 



Substituting these values, we get 



a"" CAmns/gG^-m^n^y^g^G^ (I.) 



which general expression for the deflection we may write in two 



