690 ELECTRICAL METHODS [Chap. 10 



where 



U = L cos^ ^ + N sin^ \^ + 2M sin ^ cos ^ 1 



(10-240 

 Q = L sin^ ^ + N cos^ ^ — 2M sin ^ cos ^. J 



Fo^'ming the difference D, we have 



£) = XJ - Q = (L - N) cos 2^ + 2M sin 2^, 



and substituting eq. (10-24/) we get 



2M ^ L - N 



The difference is 



Z) = U - Q = db\/4M=' + (L - N)^ (10-25a) 



Smce from eq. (10-24i) the sum U + Q = L + N, and U - Q = D, the 

 coefficients are U = (L + N + Z))/2 and Q = (L + N - D)/2. In the 

 standard form of the elUpse, (X^/A^ + Y^/B^ = 1), A is the major semiaxis 

 and B the minor semiaxis. By comparison with eq. (10-24/i) the squares 

 of the semiaxes of the elhpse in the x'y' direction are a = 1/U and 

 h^ = 1/Q, so that 



^2 2 



L + N - V'4M2 + (L - N)2 

 and 



b'= 2 



L + N + \/4M2 + (L - N)2" 



^ (10-256) 



The sign before the radical determines which is the major and which the 

 minor axis. By substituting the values forL, M, and N from eq. (10-24c), 

 the square of either semiaxis is 



a^ h' = ^ 2A^B^sin% ^^^^^^ 



B' + A' + V4A2B2 cos2 ^ + (B2 - A^)^ 



By substituting the ratio major axis/minor axis = a/h = r, and by further 

 substituting the tilt angle relation given in eq. (10-24^), formula (10-18c) 

 is obtained, allowing for the fact that the tilt angle is reckoned from the Y 

 (vertical) component. 



Two fields 90" out of phase, forming an arbitrary angle with each other 

 Ukewise give rise to elliptical polarization. The theory is treated in sec- 

 tion vni (page 787), as it is of importance in electromagnetic and inductive 

 electrical prospecting methods. 



