Determining the Angle of Lag. 371 



The equation of the long diameter of the ellipse is 



y= -x. 

 u a 



Then since the short diameter is perpendicular to this, its 

 equation is 



a 



y=-- b *. 



Treating either of these equations as simultaneous with the 

 equation of the ellipse, the coordinates of the intersection of 

 these diameters with the curve may be found, from which 

 may be deduced the lengths of the diameters in terms of <ft. 

 This result is general but too cumbersome to be of service. 



Suppose a = b, a condition which may be easily attained by 

 increasing or decreasing the current in the vibrating loops, 

 or by varying the strength of the actuating magnets. The 

 equation of the ellipse would then be 



y=xco$(f>— \/l—x 2 sin (ft, 



the equation of a family of ellipses whoso parameter is (ft. 

 The equation of the diameters is 



y=±x. 



Combining this with the equation of the curve, there re- 

 sults as the squares of the coordinates of intersection 



2 _ sin 2 (ft 2 _ sin 2 (ft 



y ~2(l + cos(ft) ; X ~2(l + cos</>)* 



The upper sign in the denominator belonging with the 

 positively sloped and the lower with the negatively sloped 

 diameter. The squared length of the semidiameters is the 

 sum of these squares, or 



sin 2 (ft 



d- 



1 + cos (ft' 



The whole diameter is double this semidiameter, so calling 

 D : the positively and D 2 the negatively sloped diameter, 



2 _ 4 sin 2 <ft 

 1 "1- cos <£' 



4 sin 2 (ft 

 U * "l + coscft- 



