282 Mr. L. Fletcher on the Dilatation of Crystals 
whence 
tan Q'T" tan (Q"X-P"X) 
tanQP == tan(QX-PX) 
^cotPX-^tanPX 1 + 
o a b 
+ b 2 u 
a + /3 a&(« + |8) 
a o 
-cotPX-tanPX 
a 
(5) 
a relation which holds rigorously for any crystal-section con- 
taining two rectangular atropic lines. 
The planes P, Q will thus not be permanently isotropic unless 
a£>(a + /3) equals ab 2 + fia 2 , or = — j for all temperatures — a 
condition which is only satisfied for those crystals in which a 
is permanently equal to /3. 
In the case of rock-crystal, according to Fizeau's measure- 
mGnts if 
^ = 10° C, * 2 =80°C., and f 3 = 50°C., 
a = 1-00100163, 
/3 = 1-000552895, 
a = 1-00055808, 
5=1-00030532; 
whence, if P, Q be two planes isotropic for the pair of tempe- 
tures 10°— 80° C, the angle between them at 50° C. will be 
given by the equation 
'TZr = ab ,t + /} =1-00000002473, 
tan QP u¥ + fia 2 
or, to the second order of small quantities, 
Q"P"-QP=0-000000012365sin2QP. . . (6) 
Hence the variation in the angle PQ will not exceed 3.-^3 th of 
a second. 
If a = l+m, /3 = l + n, a=l + d, b = 1+e, and higher 
powers than the square of the small quantities m, n, d, e be 
neglected, it will be found that 
a6(q + |8) _ 1 (d — e)(m — n-d + e) 
aft 2 + /3a 2 "* + 2 ' 
* Comptes Rendus, 1866, vol. Lai, p. 1145. 
