of Currents in a Primary and a Secondary Coil. 407 

 would be similarly given by the expression 



I 2 =Vr 2 2 + K 2 p 2 . 

 Let p 2 be denoted by x, and, for shortness, let 



a=r 1 r 2 — (LN-M. 2 )x, 



6=(Nr 1 + Lr 2 ) V5. 



Then, assuming that the E.M.F., namely e, applied to the 

 primary is at any time, t, given by the equation 



£=E sin (pt + ()>), 



the periodic parts, c x and c 2 , of the currents at this time in 

 the primary and secondary, respectively, are given by 



EI 2 . 



Ci = , sm pt, 



Va 2 + b 2 F ' 



and we have 



EMV t ? . 



C2= VPrP sm( ^ + ^ 



# = i|7r— p£, where tan ^;= , 



(j) = i|r — p£, where tan ^ = — 



It thus follows that the actual impedances, I, F, of the pri- 

 mary and the secondary coil during the working of both are 

 given by 



T2 _ a 2 + & 2 . V2 _a 2 + b 2 m 



1 ~ J 2 2 ' X " M 2 ^ w 



To represent I, I 7 , and the phase-angles 6 and <£ graphically 

 is the problem in hand. Take two rectangular axes, 0#, Oy, 

 and along the first lay off the numerical values of p 2 ; then, 

 taking k 2 y to represent the value of P corresponding to any 

 value of p (or x), where fc 2 is any constant which (according 

 to the numerical values of L, M, N, &c.) may be required 

 to confine the figure to any convenient size, we have 



k 2 y= i + *' , (2) 



or if for shortness we put A 2 = LK— M 2 , and B 2 = Nr 1 + Lr 2 , 



Py{~N 2 xi-r 2 2 ) = (r ± r 2 - A^ + B^x; . . . (3) 

 2E 2 



