428 On the Dilatation of Crystals on Change of Temperature. 
and 
log cos L'"Y, 9-0269404-6 log cos L'"Z, 9-41334453 
logcosM'"Y, 9-8940005-5 logcosM'"Z, 9-7898455"7 
log cos N'"Y, 9-7869667-7 logcos N'"Z, 9-8713669 ! 
which are almost identical with the original values given in 
§ XVIII. 
XXI. For convenience we here tabulate the above variations 
from i)0 o of the angles between the crystal-lines L, M, N at 
the different temperatures : — 
log cos L'"X, 9-9822675-1 
logcosM'"X, 8-9002634-3 
loecosN'"X, 9-4289636-6 
Variation of 
LM. 
MN. 
LN. 
it 20 C. .. 
„ 80 C. .. 
„ 140 C. .. 
„ 200 C. .. 
6 
. + 6 
. -17 


+ 16 
+ 8 


+ 32 
+ 35 

The variations at 80° C. and 140° C. of the angles between 
the three crystal-lines which are at right angles both at 
20° C. and 200° C. are thus very small ; even if parameters 
determined by experiment are accepted as perfectly accurate. 
XXII. Proceeding in exactly the same way we have cal- 
culated the positions of the lines which are at right angles 
both at 20° C. and 80° C, and find that they are virtually 
coincident with those calculated by Beckenkamp ; hence wo 
infer that he is correct in stating that the directions of the 
thermic axes as calculated from his measurements widely differ 
for the two pairs of temperatures. 
XXIII. The variations tabulated in § XXI. are, however, 
sufficient to prove that even this wide difference of about 26^° 
results from changes in the relative inclinations of LMN so 
small that they might be accounted for by the minute errors 
incidental to the measurement of the angles. In the first 
place, although single seconds were read off by the micro- 
scope, an angle, when determined with a rising temperature, 
varied in some cases from that determined with a falling 
temperature by as much as 10", the average variation being 
&''l ; secondly, the corrections derived according to the 
method of least squares and applied by Beckenkamp to the 
measured angles have an average value of 3"*2 and reach a 
positive maximum of 9"'7 and a negative maximum of — 8"-7; 
thirdly, although it is not impossibi >, ^till it is curious that 
the pair of lines LM should be at right angles not only at 
20° 0. and 200° C. but also at a third temperature between 
80° C. and 140° C; and lastly, as Beckenkamp points out in 
the later paper, there may be a minute distortion of the 
