416 Mr. L. Fletcher on the Dilatation of Crystals 
VII. As was pointed out above (p. 347), angular measurement 
of a crystal can result in the determination of only the ratios 
A : B : C and A' : W : C, and not of the absolute lengths of 
the axes. If the parametral ratios are, as usual, given in the 
form a : 1 : c and a' : 1/ : c', and ^- = 1 + X, we shall have, 
as before, 
13 
i-a«£ 
+\, 
+ \. 
, A' 
B' C 
Substituting these values of ^j ^ ? p- in the above expres- 
sions for P, Q, R, D, E, F, and still neglecting small quantities 
of the second order, it will be found that 
P=p+X, V = d, 
Q = q + \, E = e, 
R=r + \, F=f, 
where 
r 
P = 
d= 
8' 
-1. 
a sin 8 
a! cos B' cos C 
sin a! sin C 
sin a sin C 
/ cos 8 cos C 
r=--l, 
cos a 
a sin /3 sin C c sin 
a' cos/3' d cos 8 
a sin /3 c sin /3 
a' sin 
sin (J sin a sin C 
c cos a 
c sin a sin (J 
a sin a sin 
cos C sin a' cos 0' 
C sin a sin 
We have now determined the expansions and rotations of 
three crystal-lines which are initially perpendicular in terms 
of the coefficient of expansion A, of a line in the direction OB 
and of the parameters of the crystal at the two temperatures. 
VIII. Let \ fii Vi, A, 2 ^2 v 2i ^3 ^3 v 3 De the direction-cosines 
of the lines OX/, OY', OZ', referred to the three rectangular 
axes OX, OY, OZ ; OX 7 , OY', OZ' being the positions at the 
second temperature of the crystal-lines which coincide with 
OX, OY, OZ at the first. Then, neglecting squares of small 
quantities, 
or OX', 
*i = l, 
/*i=0, 
v x = E 
for OY', 
\ 2 = F=f, 
^2 = 1; 
v 2 = D 
for OZ', 
X 3 = 0, 
^3=0, 
v 3 = l. 
