168 BELL SYSTEM TECHNICAL JOURNAL 



^22 = fi^n + 2i2nhl3n + ^(^fh^n + ^2^22 + ^m^th^^z + nl^zz 



/323 = (iCz^n + (4w3 + W2^3)iSi2 + (/2W3 + ^Jz)l3n + W2W31822 



+ (W2W3 + n2mz)^23 + rhnz^zz 



/333 = 4/3u + 2Czmil3i2 + 2fznz^n + W3/?22 + 2mznz^2z + ^3^33. 



Now, if the axes refer to the axes of a Fresnel eUipsoid, /3i2 = /3i3 = 1823 = 

 and one of the impermeability constants for any direction, say 1833 , can be 

 expressed in the form 



/333 = (l^i + ^3^2 + nl^z (32) 



If r, which hes along Z' of Fig. 2, is the radius vector of the Fresnel ellipsoid, 

 then the direction cosines 4 , niz and M3 are 



f X y z 



^3 = -, mz = -, nz = -. 

 r r r 



From equation (24) /3i = a''/V\ ^2 = byV\ jSj = cVF^ and equation (32) 

 becomes 



2Tr2o' 2 2 I ,2 2 I 2 2 ^ 



Hence the square of the radius vector of the Fresnel ellipsoid is 1/F"j833 

 and the radius vector of the impermeability ellipsoid agrees with that of the 

 Fresnel ellipsoid. Hence, the directions of vibration can be determined from 

 the principal axes of the impermeability ellipsoid for any diametral plane. 



When light transmission occurs along Z', the direction for maximum and 

 minimum impermeability can be obtained by evaluating jSn and deter- 

 mining the angle xp for which it has an extreme value. Inserting the direction 

 cosines d , mi and Wi from equation (27), we find 



o' o r 2„ 2 2 , sin 2^ sin 21/^ cos ^ , .2 • 2 ."^ 

 Pi\ = Pi cos 6 cos ^ cos xp — -~- + sm ^ sm xp 



, Q \ 2 /J • 2 2 , , sin 2^7 sin 2i/' COS , 2 . 2 , 

 + 182 cos 6 sm ^ cos \p -\- — —^ [- cos (p sm \p 



+ 183 sin" 6 cos" yp. 



03) 



Differentiating with respect to \p and setting the resultant derivative equal 

 to zero, the value of }p that will satisfy the equation is given by 



o / (182 — (3i) sin 2<p cos d 



tan 2\p = 



(/3i - 182) (cos2 d cos2 <p - sin2 <p) + (^83 - ^2) sin^ 6 



{b — a) sin 2<p cos 6 

 (^2 - 62) (cos2 d cos2 <p - sin2 <p) + {c^ - b^) sin^ d ' 



