284 Mr. L. Fletcher on the Dilatation of Crystals 
the relative positions of the poles if arcs measured in opposite 
directions are regarded as opposite in sign. 
Prop. I. Given the alterations of the angles between three 
planes of a zone, to determine the displacement of a fourth plane 
of the zone [see also Props. V. and VI.] (fig. 2). 
Let the poles a, c, d at the first temperature take up the posi- 
tions a',c',d' at the second; and let it be required to find P', 
the position at the second temperature of a pole having the 
position P at the first. 
From the constancy of the anharmonic ratios we have 
[cVPV] = [cdPa], 
and 
whence 
If 
-in 
P V sin d'c sin Pa sin dc 
sin PV sin d'a! sin Pc sin da 
sin PV sin Pa sin dc sin d'a 1 
sin (PV — cV) sin Pc sin d'c 1 sin da 
sin Pa sin dc sin d'a' 
tamlr — 
T sin re sin dc sin da 
a known quantity, then 
tan (PV- ~) = tan ^ cot (^-45°), 
a logarithmic formula by means of which the arc PV can 
readily be calculated. 
Prop. II. Given the alterations of the angles between three 
planes of a zone, to determine the position of a plane in the same 
zone isotropic to one of them. 
As before, let a, c, d be the given poles, and let it be required 
to find a pole a isotropic to a. 
From the constancy of the anharmonic ratios, 
[acda~\ = [a'c'd'a'], 
whence 
sin ac sin da _ sin aV sin d'a' 
sin a a sin dc sin a! a! sin d'c' 
By hypothesis, 
aa = a'a' } 
and thus 
sinac sindc sindV 
sin a V sin da sin dV 
sin (ac—aa) _ sindc sindV 
sin (a V — ad) sin da sin d'c' 
