568 Dr. S. P. Thompson and Miles Walker on the 



Let the impressed volts be called V, the current i (both 

 being square-root of mean square values), the resistance of 

 the coil R, its coefficient of self-induction L, the frequency of 

 the period of alternation n, and the number of windings W. 

 The inductance will then be 27rnL, which for brevity may be 

 written ph. Then the impedance will be \/W i +p 2 L 2 . The 

 relation between volts and amperes in the coil will be expressed 



as Y=i y/R*+p*L*. 



The angle of Jag being called <£, its sine will have the value 



pL-r- */R 2 +p 2 IA 



Hence we have the following relation : — 



Y . sin <j) = iph ; (1) 



these being equivalent expressions for the effective part of the 

 volts that is employed to balance the counter-electromotive 

 force of self-induction in the coils. Now L is proportional to 

 the square of the number of windings ; and may be written 

 = kW 2 , where k is the coefficient of self-induction of a single 

 turn, and W the number of windings. 



Now, let the prescribed number of ampere-turns that are 

 to be produced be called Z = ?W. Inserting these values in 

 (1) we obtain the expression 



Ysin0 = /fepWZ; (2) 



and finally w _ Y sin , . 



n ~ ~kpz~ ;•••••• W 



which is the relation sought. 



4. The quantity k denoted above, the coefficient of self-in- 

 duction of a single turn, depends upon the dimensions of the 

 iron parts of the magnetic circuit and upon those of the air- 

 gap, as well as upon the permeability of the iron. If the 

 magnetization is not carried beyond 5000 lines per square 

 centimetre, the permeability may be considered as constant. 

 But k will decrease with any increase in the reluctance of 

 the magnetic circuit, such as the lengthening of the air- 

 gap. On the contrary, as the armature is attracted up 

 toward the poles k increases. The quantity k represents the 

 number of magnetic lines ( x 10 -9 ) which permeate the mag- 

 netic circuit when one ampere circulates in a single turn of 

 wire around the core. We have only to multiply k by the 

 square of the number of windings to obtain the coefficient of 

 self-induction of any coil subsequently to be wound upon this 

 magnetic circuit. The quantity k must itself be experimentally 



