142- Dr. 0. J. Lodge on Intermittent Currents 



throws light upon this, and reconciles the two observations. 

 It turns out (as might have been anticipated) that Professor 

 Hughes's statement is quite correct when the coils are near 

 together ; but if they are separated by a distance greater than 

 the diameter of either (the coin being supposed small), the 

 middle point becomes a minimum with a maximum on each 

 side of it (see next section). 



On the Law according to which the disturbance produced by a 

 small coin on the common axis of a pair of coils in an induc- 

 tion-balance depends on its position. 



19. The general expression for the induction-coefficient be- 

 tween two circular coils in any position whatever is given in 

 ' Maxwell/ art. 696. We can specialize this to the case 

 required, viz. a coil of n turns and a "coin" or coil of one 

 turn, both on the same axis. Let the mean radius of the 

 "coil" be a, and let the distance of its circumference from 

 some point on the axis be c. Also let the mean radius of 

 the " coin " be b, and let d be the distance of its centre from 

 the point 0. Then the mutual induction-coefficient between 

 the coil and the coin is (art. 699) 



2n7r 2 ^|l + 3 ^( c2 ~ a2) d + 6 c2 ~f a2 (#-^) + & . j. 



Now d we can make zero at once by taking the point at the. 

 middle of the coin, so that c is the distance from the mean cir- 

 cumference of the coil to the middle of the coin ; and there 

 remain inside the brackets of the above expression 1 + terms 

 involving the square and higher powers of b : c, which is a 

 small quantity. Thus a very good approximation to the in- 

 duction-coefficient between the primary and a small coin is 



27rWn ; 



P= — ? — • • (43) 



Similarly the induction-coefficient between the secondary 

 and the coin is the same expression with dashed letters ; hence 

 the product 



^-—{WVaa'f (44) 



And this is a maximum when cc' is a minimum, a thing which 

 is very easily represented geometrically. 



For let A and A' be points on the mean circumferences of 

 the coils from which c and c' are measured; draw the system of 

 lemniscates cc' = various constants, and then draw through the 

 system any straight line at a distance a from A and a' from A'. 



