72 lloyal lastitvtion. 



To pursue this nmtter further. Suppose that, the arrangements 

 remaining otherwise as before, the analyzer be turned round ; then 

 in any position intermediate to 0° and 90° the rings will be con- 

 tracted and extended in opposite quadrants, until at 45° they are 

 divided by two diagonals, on each side of which the colours are 

 complementary. Beyond 45° the rings begin to coalesce, until at 

 90° the four quadrants coincide again. During this movement the 

 centre has changed from bright to dark. If the motion of the 

 analyzer be reversed, the quadrants which before contracted, now 

 expand, and vice versa. Again, if the crystal (say positive) be re- 

 placed by another (say negative), the effect on the quadrants of the 

 rings will be reversed. This method of examination therefore 

 affords a test of the character, positive or negative, of a crystal. 



A similar process applies to biaxial crystals ; but in this case the 

 diagonals interrupting the rings are replaced by a pair of rectangular 

 hyperbolas, on either side of which the rings expand or contract ; 

 and the effect is reversed either by reversing the motion of the 

 analyzer, or by replacing a positive by a negative crystal, or vice 

 versd. The experiment may then be made in biaxial crystals by 

 turning the analyzer slightly to the right or to the left,. and observing 

 whether the rings advance towards, or recede from, one another in 

 the centre of the field. In particular, if, polarizer and analyzer 

 being parallel, the plate A have its axis in a N.E. direction to a. 

 person looking through the analyzer, the plate B its axis in a N.W, 

 direction, and the crystal be so placed that the line joining the optic 

 axes be N.S., then on turning the analyzer to the right the rings 

 will advance to one another if the crystal be negative, and recede if 

 it be positive. The mathematical expression for the intensity of the 

 light at any point P is in this case 



1(1 + sin 2y' cos Q + sin '2h cos 2/' sin 0), 

 where h is the angle between the principal section of C through P 

 and the principal section of B, and J the angle between the principal 

 sections of B and the analyzer. This shows that when the polarizer 

 and analyzer are parallel or crossed at 0° or 90°, and consequently 

 j = 45° or 135°, the expression is independent of b (i. e. the intensity 

 is the same throughout circles about the centre), but that when the 

 polarizer and analyzer are crossed we have an expression of the form 

 J(l + sin 2i sine), 



the sign of the second term depending upon the direction in which 

 the analyzer has been turned, and also upon the sign of — that is, 

 upon the character (positive or negative) of the crystal. 



The dispersion of the planes of polarization effected by the passage 

 of plane-polarized light through a plate of quartz cut perpendicular 

 to the axis may be rendered visible by interposing such a plate of 

 quartz between the polarizer and a uniaxial or biaxial crystal when 

 the analyzer is at 90°, i. e. when dark brushes are formed. In this 

 case the brushes cease to be black and are tinged throughout with 

 colour. The analyzer, however, must be turned back or forward, 

 according as the quartz be right-handed or left-handed, in order that 



