the Double Refraction of Quartz. 465 
Again, since a?—b’=.00489, we have 
Lipa 
——_ (a? —b”) = 258, 
ae 
so that equation (8.) becomes 
k? — 258 sin’ ¢. k=1. (13.) 
The results of this formula are compared with Mr. Airy’s experiments in the fol- 
lowing table, in which the less root is taken for /, and its sign is neglected. 
Values k. | 
Values of ¢. - 
Observed. “| Galcalated. | 
6° 15’ | tan 16° 38'=.2987 | .2980 
8° 54 tan 8° 56 =.1572 1579 
The angles 4, in the first column, are deduced from the observed inclinations of the 
rays in air to the axis of the crystal; and as &/ was observed to be somewhat different 
for the ordinary and extraordinary rays, its mean values are given in the second 
column. ‘The exact coincidence between these and the calculated values is, perhaps, 
in some degree accidental ; but a less perfect agreement would be sufficient to confirm 
the theory. 
The magnitude of k varies considerably with the colour of the light, increasing 
from the red to the violet, while the coefficient of sin? ¢. k, in formula (18.) diminishes. 
If we take the violet ray H, for example, this coefficient will be about 159. But 
it would be useless to make any more calculations, as we haye no experiments with 
which they might be compared. 
The figure of the wave surface yet remains to be examined. 
Eliminating k between formulz (5.) and (6.), we obtain the equation 
(#— A)(—B) =4n? <, 14.) 
which expresses the nature of the surface, s being a perpendicular from the origin on 
a tangent plane. From this equation it follows that the two values of s can never 
become equal in quartz, as they do in other crystals ; for if we solve the equation for 
s’, and put the radical equal to zero, we shall get the condition 
(A-B)'+160"S =0, 
which cannot be fulfilled, since the quantity which ought to vanish is the sum of two 
squares. The two sheets, or nappes, of the wave surface, are therefore absolutely 
separated. 
