on Change of Temperature. 347 
For the line perpendicular to OP, where ^ , i = ^ + ^-, we 
have 
"^i' - ^1 — — (B,— P) sini/r cos-^ + E sin 2 ^, 
whence, if the lines be isotropic, we have, by equating the above 
values, 
tan2^ = p^ (VI.) 
Second Method. — From equations (T.) it is seen that the 
circle a? + z 2 = l will become the ellipse 
(l-2P)f-2E^+(l-2R)$ 2 = l, 
whence, by the usual formula, if yjr, i/r + - be the inclinations 
of the axes of the ellipse to the axis OX, 
F 
tan2^r=p- 1 , 
as before. 
(7) By measurement of the angles of a crystal, not the abso- 
lute lengths A, B, C, but the ratios A : B : C are determined, 
and are generally expressed by the symbol a : 1 : c. If the 
parametral ratios for the two temperatures be a : 1 : c and 
a : 1 : c', we shall thus have A = Ba, C = Bc; similarly for the 
second temperature, A' = B'a', C' = B'c' ; whence, if X be the 
coefficient of expansion in the direction OB and therefore 
B' 
-p=(l+\), we may write, still neglecting squares of small 
quantities, 
f'= ( i + x)f'=-: + >, 
C v J c c 
A! C 
Substituting these values of -r-, ~- in equations (III.), we find 
that we may write 
F=p + \, R=r + X, E = e, "| 
where 
_ a' sin /3' 
" ~~ a sin /3 
(VII.) 
r = 
c 
-h 
e- 
a 
cos/3 7 
sin/3 
d cos j3 
c sin j3 
