on Change of Temperature. 285 
' .' 
If , sin dc sin d r a 
tan 9 
sine^V sin. da 
a known quantity, then 
(a'cf + ac \ a!c' — ac, /aKQ , , N 
tan ( k act )= tan — ^ tan (4o + <p), 
from which the arc aa can be readily found. 
Peop. III. To find a relation beticeen any two pairs of iso- 
tropic planes in the same zone. 
If a, a, P, Q be two pairs of isotropic planes, we have, as 
before. 
[aPQ«] = [a'P'QV], 
or 
And since 
we have 
sin aP sin Qa _ sin a'V sin QV 
sin act sin QP ~~ sin a' a! sin Q / P / 
aa=aV and QP=QT / , 
sin a'V sin Qa 
or 
But 
hence 
sin aP sin QV 
sin (aP + PP'- aa') _ sin (QV + QQ'- 
sin aP sin QV 
PP'=QQ'aiidaa / =a«'; 
aP = QV. 
-aa.') 
Or, if two pairs of planes be simultaneously isotropic, the angle 
between two of the planes, one from each pair, at the first 
temperature is equal to the angle between the other two planes 
at the second temperature. 
Corollary. — In the first paper it was shown from mecha- 
nical considerations that, for any pair of temperatures, at least 
one pair of real planes is atropic. 
If a, a be the poles of a pair of these planes, the relation just 
given becomes 
aP=QV 
Or if in any crystal two planes are isotropic for a given change 
of temperature, the angle which one of the isotropic planes 
makes at the first temperature with either of the atropic planes 
is equal to the angle which the second isotropic plane makes 
at the final temperature with the remaining atropic plane. 
