Chap. 10] ELECTRICAL METHODS 689 



which has the form 



Lrc' + 2Mxy + Ny' = 1 (10-24c) 



and is the expression for an inclined elUpse whose major axis is tilted in 

 reference to the x axis. In this equation, 



A2sin2^' ABsin^v?' B^sin^^* 



If the phase shift is 90°, it is seen that the major axis of the ellipse coin- 

 cides with the X axis, so that 



A2 + B3 = 1' (10-24^) 



which is the standard form of the ellipse. To determine the angle 

 of deviation xp of the ellipse (given by eq. [10-24c]) from the x axis, ro- 

 tate the system of coordinates so that x = Xi cos ^ — yi sin ^ and 

 y =Xi sin rp -{- yi cos \f/. 

 By substitution in (10-24c), 



xl[L cos'^ ^ + N sin^ \l/ + 2M sin }{/ cos \p] 



-\- yl[L sin^ i/' + N cos^ i^ - 2M sin xf/ cos i/'] \ (10-24e) 



+ 2xiyi[{JX — L) sin i^ cos i^ -f M(cos^ xf/ — sin^ yp)] = 1., 



In this new system an equation of the form of (10-24rf) must obtain, 

 and therefore the coefficient of 2xiyi must be zero. This leads to 



2M 



tan2^ = j-^^. (10-24/) 



By substituting for L, M, and N their values given before, 



ltan2,A = ^^, (10-24g) 



B ~ A 



which is the same as eq. (10-18d) given in connection with the discussion 

 of the elliptical polarization of radio waves (there the tilt angle was 

 measured from the Y and not from the X axis). 



The axes of the ellipse may be determined from eq. (10-24e) by casting 

 it in the form of the standard ellipse as in eq. (10-24d). Designating the 

 coefficient of xl by U and that of yi by Q, 



Vxl -1- Qyl = 1, (lO-24/i) 



