278 Mr. L. Fletcher on the Dilatation of Crystals 
proved. An examination undertaken with this object in view 
has led to the gradual development of the present paper. 
It was soon found that the formulae of Neumann are so 
complex, and are arrived at by so complicated a method, that 
it was far from easy to become quite confident that some small 
error might not have crept in to invalidate the result, more 
especially as Beckenkamp had himself had occasion to point 
out two printer's errors in the formulae of the original paper of 
Neumann. This feeling will perhaps be more intelligible if we 
give Neumann's statement of the method by which the ther- 
mic axes are to be actually calculated. It is as follows : — 
" Let OA, OB, OC be the crystal! ographic axes and have 
lengths A, B, C respectively: take three rectangular axes x h .r 2 , 
# 3 , # 3 being coincident with OC, x 2 in the plane AOC and 
perpendicular to OC, and a^ perpendicular to the plane AOC. 
Let a x a 2 « 3 , /3 X j3 2 /3 3 , <y x <y 2 73 be the direction-cosines of OA, 
OB, OC at the initial temperature 6 with respect to the axes 
#n#2)#3 5 l e t the increments at the second temperature 6 of 
the nine cosines u 1} « 2 , «3> &c- &c. be denoted bv Aa x , Aa 2 , . . . , 
and of the lengths A, B, C by AA, AB, AC. For brevity 
write 
AA 
A 
AC Aa . AB AC Ab 
and let a denote the determinant 
a l «2 a 3 
Hi H2 P3 
7i 72 7s 
Find nine constants A from the following formula :- 
da . da . n da . 
d*k dfa r dy k da k 
da Aa da Ab „ 
and construct the equations 
2(A/-R)C 1 + (A/' + A 2 ')C 2 + (A^+A.0 C 3 =0, 
(A/ + A/0C 1 + 2(A 2 ' / -E)C 2 + (A 2 "' + A 3 ")C 3 =0, 
(A 3 ' + A 1 /// )Ci+ (A 3 " + A 2 "')C 2 + 2(A 3 '"-R)C 3 =0, 
C? + 
C + 
CJ =1. 
Determine the roots R a , R 2 , R 3 of the cubic equation 
2(A/-R) (A/' + A/) (A^+A.0 
(A/ + A/') 2(A 2 "-R) (A 2 "' + A 3 ") 
(A 3 ' + A/") (A 3 "+A 8 "') 2(A 3 '"-R) 
