on Change of Temperature. 279 
Substituting in turn these roots in the preceding equations, 
calculate the values of C 1} C 2 , C 3 : each set of values of C^^Cg 
will give the direction-cosines relative to the rectangular axes 
asi i % 2 , x 3 of the directions at the second temperature 6' of 
thermic axes for the pair of temperatures 6 and 9' '." 
It is not recorded that Neumann himself made any practi- 
cal use of these formula?; and it would appear that Pape, by his 
determination of the thermic axes of crystals of copper sul- 
phate*, was the first to show that they come within the range 
of practical crystallography : the actual results, however, can- 
not be very precise, seeing that the instrument employed only 
afforded readings to a minute. Beckenkamp has now made 
extensive use of these formulae to ascertain whether or not in 
an Anorthic crystal the same crystal-lines remain permanently 
at right angles, and decides the question in the negative. 
It had occurred to the writer that results deduced from these 
formulas would really be useless for the purpose of testing the 
permanency of the thermic axes, since the equations are ob- 
tained by neglecting small quantities of the second order upon 
which one would expect any change in the thermic axes to 
depend. In fact it will be shown later that on reversing the 
order of the temperatures, exactly the same equations, and con- 
sequently the same direction-cosines, as before will be ob- 
tained; but this time they must give the position of the thermic 
axes at the lower instead of at the higher temperature. In 
other words, the formulas only give the position of the thermic 
axes within an angular distance equal to the actual displace- 
ment of the corresponding crystal-lines by the change of tem- 
perature. 
As, however, the calculated deviation of 26° 36' in the position 
of one of the thermic axes of anorthite is evidently far beyond 
the displacement of any lines of the crystal, it would appear to 
result from some other cause than the neglect of the terms of 
the second order. 
Since it is very difficult to imagine that two lines of a 
crystal can have the same mutual inclination at two different 
temperatures and yet deviate therefrom at intermediate ones, 
we propose to show in the first place by actual calculation 
that such a property is undoubtedly characteristic of crystal- 
line bodies ; and incidentally we shall find the deviation in 
the case of the particular mineral used for illustration, thus 
obtaining a clue, however faint, to the order of magnitude 
of this deviation in the general case. 
Ann. 1868, to!, cxxxv. p. 1. 
