426 Mr. L. Fletcher on the Dilatation of Crystals 
whence 
log cos BX, 8-7065376-2; log cos BY, 9-9987849-7; 
log cos BZ, 8-7379172-1. 
From the logarithms given in (a) and (6) we find 
Beckenkamp. 
BC . . 
. 93 8 6-5 
93 8 6-6 
CA . . 
. 115 53 8-4 
115 53 8-4 
AB . . 
. 91 15 17-0 
91 15 17-1 
a result which shows that the formulas of § XIII. are so far 
perfectly satisfactory. 
XX. We now proceed to determine the relative positions 
L M N, 1/ M' W, L" M" N" of the lines L'" W" N'" at 20° C, 
80° C, and 140° C. respectively. 
(a) To find the positions L M N at 20° C. 
If OP'", £77 £, and OP, sey z be positions of the same crystal- 
line at 20° C. and 200° C. respectively, then, from § XIX. (b), 
- =0-9982745^ + -0004525, ^) 
y v ' ! 
-=0-9999435-? --0010007^ - -0001257. \ 
y v v J 
Substituting in these formulas the values of tho Jirection- 
cosines given in § XVIII., we deduce 
log cos LX, 9-0823151 -8 I log cos LY, 9-0277586-8 
log cos MX, 8-8976047-8 log cos MY, 9-8940320 2 
log cos NX, 9-4285864-8 | log cos NY, 9-7868904-4 
whence 
LM=90° + 0"-08, MN=90° + 0"-02, NL = 90°4 0"'18: 
thus confirming the statement that the above lines are virtually 
at right angles both at 20° C. and 200° C. 
(b) To find the positions 1/ M' W at 80° C. 
For 20°— 80° C, according to § XV., 
p-q= + -0004452, /=--0000376, 
,,_^ = _.0000845, e=+-0001522, a 7 = -0000419; 
and we have from § XII., if P' be £??£, and P be xyz, as before, 
i = (i +p - q yl + f, 
v y 
y z X 
z=(l+r-q)~ +e- +d, 
v v y y 
log cos LZ, 9-4125499-3 
log cos MZ, 9-7S98387-6 
log cos NZ, 9-8714577-9 
