of Currents in a Primary and a Secondary Coil. 409 



whose ordinate we shall denote by h, so that 



B 4 



h= ¥W ( 6 ) 



It is easily seen that both hyperbolas cut the axis of x in the 

 same two points. These points are at the left of and not 

 shown, since negative values of x (i. e. p 2 ) do not belong to 

 the physical problem. Moreover, the x of H is the geo- 

 metric mean between the intercepts of the hyperbola on Ox. 



Again, the centre of the hyperbola (6) is at the point C 

 whose coordinates are 



n , NW + LV + 2MV 2 



u ana ww ; 



while one asymptote is Oy and the other is CS whose direc- 

 tion is easily known, since the tangent of its inclination to 



n ■ AL 



1S k 2 M. 2 ' 



Hence we have at once the asymptotes, Cy, CS and one 

 point, H, of the hyperbola, from which the curve is rapidly 

 drawn by the well-known rule that the parts intercepted 

 between the curve and its asymptotes on every line drawn 

 through H are equal. The other branch of this hyperbola is 

 not represented, as it is irrelevant. 



We shall now show that the primary hyperbola can be 

 drawn from the secondary. Representing the values of I x 2 

 and I 2 2 for all values of x by ordinates, so that k 2 y i = Ii 2 , 

 Py 2 = I 2 2 , we see that 



Fy^IAe + V, ...... (7) 



Py 2 =Wx+r 2 2 ; (8) 



so that the impedances are now represented by two right 

 lines, AL and A 7 !/. (These are the impedances of the coils, 

 each treated separately, as before explained.) 



The primary line (7) passes, of course, through A, and 

 always intersects the primary hyperbola in a point, Q, having 

 a positive abscissa, viz., 



2LN-M 2 ' 



which is < twice abscissa of H. Hence for some speed less than 



that represented by the abscissa of Q the ratio j- attains a 



maximum value. The point, p, representing this maximum 

 value is easily found ; for, no matter what curve AQP may 

 be, if y is the ordinate of a point on it, and y ± the correspond- 



ing ordinate of the right line AQ, the ratio — is a maximum 



