288 Mr. L. Fletcher on the Dilatation of Crystals 
or minimum, 
2cos2Pasin 2 (Pa + a) + 2sin2Pasin(Pa + a)cos(Pa + a) = 0, 
or 
sin (Pa + a) sin (3Pa + a) = 0. 
If sin (Pa + a) = the displacement of the pole is absolutely 
zero. 
From sin(3PaH-a) = we deduce 
Pa=^ ^+60°, or ^+120°. 
The corresponding values of the above term are 
F . 3 2 P . 3 /2aa , 10A0 \ P . 3 /2«a 1QAO \ 
g-sin^aa, ¥ sm»(-3-+120°J, ^ sm^_-120 o j. 
Prop. IX. To find the lines of greatest and least expansion 
lying in a given zone of any crystal. 
If 6 be the common displacement of any two isotropic poles 
P, Q lying in the zone, we find from Prop. V. that 
cot P« - (1 + e) cot Pa cot Qa — (1 + e) cot Qa' 
whence 
sin Pa sin Qa 
sin Pa sin Qa ~~ ' 
an equation which may readily be transformed into a loga- 
rithmic formula for determining Q the plane isotropic to P, 
if a, a, e, and P be given. 
If the isotropic planes be rectangular, as is the case when 
their intersections with the zone-plane are the lines of greatest 
and least expansion, 
Qa=Pa + 90°, 
Qa=Pa + 90°; 
and the above relation becomes 
sin 2 Pa ., 
• op =l + e, 
sin 2 Pa 
whence 
tan (2Pa— aa)= ( 1 + J tan aa, 
from which two positions of P may be deduced. 
Prop. X. If the atropic planes be permanerit and two other 
planes be isotropic for a given pair of temperatures, to find the 
change in the mutual inclination of the latter at other tempe- 
ratures. 
If 6, <f> be the displacements of P and Q respectively for a 
