424 Mr. L. Fletcher on the Dilatation of Crystals 
whence 
logcosN'"X. 
(1.2) 9-4289315-4 
(2.3) 9-4289430-7 
(3,1) 9-4289274-6 
And for mean, 
(1,2,3) 9-4289340-2 
logcosN"Y. 
9-7869912-9 
9-7869958-2 
9-7870116*5 
9-7869995-9 
logcosN'"Z. 
9-8713444-9 
9-8713399-5 
9-8713313-4 
9-8713385-9. 
It will be found by actual calculation that, in each of the 
three cases, the sum of the squares of the cosines corre- 
sponding to the above mean logarithms is unity. 
The angles L'"M'", M'"N'", N'"L'" calculated from these 
mean logarithms will be found to be respectively 90° +6", 
90° + 9", 90° -3", and thus still differ from 90°. 
From the fact that the values determined from the different 
pairs of equations present in each case only small differences, 
we conclude that the above mean direction-cosines cannot be 
very far from precise; and, further, that it will be as laborious 
as useless to attempt to obtain a nearer approximation by the 
help of seven-figure logarithm-tables. We shall therefore now 
content ourselves, after thus proving that the differences are 
accounted for by ihe neglect of small quantities of the second 
order, with finding a set of lines exactly perpendicular to each 
other, and as nearly as possible coincident with the above 
positions of L'", M"', N"\ 
It will be found that such a triad of lines is defined by the 
numbers given below: — 
log cos L'" X 9-9822675-4 log cos I/" Y 90269392-4 log cos L'" Z 94133462-5 
log cos M'"X S-9002688-0 log cos M"Y9'89400130 log cos M'"Z 9-7898459-4 
log cos W"X 9-4289635-6 log cos N'"Y 9-78696617 log cos N'"Z9'8713562-8 
And we shall henceforth ndopt these lines as the positions at 
200° C. of the lines which are at right angles both at 20° C. and 
200° C. As has been pointed out above, they may, to the same 
degree of approximation, be regarded as the positions at 20° C. 
of the lines which are at right angles at the same pair of 
temperatures. 
XIX. Before we proceed to determine the variation of the 
angles between these lines atthe several temperatures, we shall 
put on their trial the formulas to be used. For this purpose 
we shall calculate, with the help of the values of^, q, r, d, e,f 
given in § XV., the angles between the crystaUographic axes 
at 20° C. from the angles at 200° C, and compare them with the 
values given by Beckenkamp. 
