and the Theory of the Induction-balance. 



143 



This line will represent the common axis of the two coils, and 

 points on it represent possible positions of the coin. Now 

 the line cuts every lemniscate in four points: hence there are 

 four points on the axis at which the coin produces the same 

 disturbance. These can run together two and two at points 

 where the line is touched by the lemniscate; and these are the 

 maximum or minimum points. They are maxima if the line 

 is touched on the side of A A', but minima if the line is 

 touched on the other side. 



If the coils are the same size (i. e. if a = a f ), and if they are 

 at a proper distance apart (viz. = 2a), all four points can run 

 together in the middle, which is therefore then a stationary 

 point, so that moving the coin a little either way will produce 

 very little diference. But if the coils are closer together than 

 this, two of the points of section become imaginary, and hence 

 there are now only two positions of the coin which give the 

 same noise ; and these two can run together in the middle, 

 which is now a maximum. 



v > 



Max. Mux. Max. 



Stationary. 



Max. 



Analytically the problem might be stated thus : — Find the 

 max. and min. values of sin 6 sin 6 f , given the condition 

 a cot 6 + a! cot Q' — const., 6 and 6 f being the angles subtended 

 by mean radii of the primary and secondary coils at the coin. 

 If the coils are equal (d=</), the solutions are 6 — & f and 

 6 + 6 f =z\ir\ hence another way of stating the result is the 

 following : — Draw a circle with A A / (the mean points of the 

 equal coils) as diameter : then if this circle cuts the common 

 axis of the coils, the points of section are maxima, and the 

 middle point is a minimum ; but if the circle does not cut the 

 axis, the middle point is a maximum, and the only one (see 

 figure). 



The law of decrease with distance is interesting : equation 

 (44) shows us that when the two coils are close together and 

 the coin is moved along the axis away from them, the effect 

 which it is able to produce in the telephone varies pretty nearly 

 as the inverse sixth power of the distance as soon as it has 

 got a little way — a tremendously rapid rate of decrease. 



It also shows that the effect varies directly with the fourth 

 power of the diameter of the coin. 



