186 HENRY A. EOWLAND 



distance in this plane from the centre of the coil we are calculating. 

 Then 



dV ' , ( n 

 - 1 



for a single circle. 

 From(l) 



and /^a--l; r' = - , 



where // = cos (9 , 



p _ 



- 



For a circle of rectangular section we must obtain the mean value of 

 this quantity throughout the section of the coil. 



1 fxo + lr, /po+H 



M=- r I I Pdxdp, 



r lZ t/x lr, t/Po-H 



where X Q and [) are the values of x and f> at the centre of section and 

 27 and c are the width and depth of the groove in which the coil is 

 wound. We can calculate this quantity best by the formula of Maxwell 

 (Electricity, Art. 700), 



Thus we finally find 



M= ^A t {l + T V + } A tll rl Q' tll + i (5, - 3) 



etc. 



It is by aid of this equation that we find the coefficients A t , A lu , 

 etc. in the expansion of the magnetic potential, V. For, let the coil 

 be moved in the field from a position where M has the value M' to 

 where it has the value M " : then if the coil be joined to a galvanometer 

 the current induced will be equal to 



M' - M" 

 R 



where R is the resistance of the circuit. If an earth inductor is in- 

 cluded in the circuit whose integral area is E, when it is reversed the 



2 J-fW 

 current is ^- where H is the component of the earth's magnetism 



