290 Mr. L. Fletcher on the Dilatation of Crystals 
with respect to a and a, it is seen immediately that any two 
poles equidistant respectively from a and a and lying in the 
arc aa are isotropic. 
(d) As a particular case, the directions of greatest and least 
expansion are inclined at an angle of 45° on opposite sides of 
the bisector of the arc aa, and thus have fixed directions in 
space at all temperatures if the planes a a are themselves per- 
manent in direction : as the crystal-lines momentarily coinci- 
dent with them are in motion, the crystal-lines cannot retain 
this property of being directions of greatest and least expan- 
sion (see previous Paper). 
(e) If 6, <j> be the variations of the arcs Pa, Qa for the same 
change of temperature, 
6 __ sin Pa sin Pa 
<f> sin Qa sin Qa 
Hence if the. planes a a be permanently isotropic, the incre- 
ment of any arc Pa bears a constant ratio to that of any 
other arc Qa for all temperatures. 
(/) If P,p be any two rectangular planes and 6, 6' the re- 
spective variations of the arcs Pa, pa, 
6 + 6' = k [sin Pa sin Pa + sin pa sin pa~\ =k cos aa, 
or 6 + 0' is constant for all positions of P and p. 
If (j>, <$>' be the corresponding variations for two other rectan- 
gular planes Q, q, then, as before, <£ + (f>' = k cos aa, whence 
6 — <fi= — (6'—<f)'); or the increase of the angle between any 
two planes P, Q is equal to the diminution of the angle between 
the planes perpendicular to them. 
Prop. XII. The motion of the plane-normal OP is quite 
distinct from that of the crystal-line OP (fig. 4). 
It is easy to fall into the mistake of assuming that the for- 
mula 6 = h sin Pa sin Pa gives the displacement of the crystal- 
line OP. That this is not actually the case will be clear from 
the following : — 
Let the pole P take at the second temperature the position 
P' on the circle of projection, and the point P the position p' on 
the ellipse : the normal OP thus becomes the normal OP', 
but the crystal-line OP becomes the crystal-line Op'; further, 
the plane tangent to the circle at P, and thus having OP for 
normal, will become the tangent plane to the ellipse at p' f to 
which the ray Op 1 is only normal when p' is at the extremity 
of a principal axis. 
The difference of motion of the plane-normal OP and the 
crystal-line OP may be found thus : — 
'Let Q be a pole distant 90° from P, so that Qa = Pa + 90° 
and Qa = Pa + 90° : if Q' be the displaced position of the pole 
