422 Mr. L. Fletcher on the Dilatation of Crystals 
90-78"-01, 90° + 73"'86 for L'"M"', M'"N'", N'"L"' respec- 
tively instead of 90°. 
As the variations of the angles between these same lines at 
80° C. and 140° C. was expected to be only a few seconds, it was 
necessary to find out to what the above differences are due, 
and for this it was practically the more convenient course to 
recalculate the axes of the quadric; the conclusion being that 
the lines given by Beckenkamp, though not quite at right 
angles, are close approximations to the thermic axes, differing 
only therefrom by small angles, probably due to the fact that, 
as the roots of the cubic equation are only approximate, the 
axial equations are, on the substitution of each of the values of 
r, only approximately consistent with each other. Indeed, 
there is considerable difficulty in obtaining the final angles 
accurate to seconds with the aid of seven-figure logarithms. 
As, however, the quadric itself is only obtained by neglecting 
squares of small quantities, such precision is really unneces- 
sary, and it will be quite sufficient for our purpose to calculate 
the positions of three lines exactly at right angles to each 
other, and as near as possible to the lines just determined. 
For the convenience of future verification, we give the more 
important numbers obtained in this calculation. 
XVIII. The equation to the 20°— 200° C. quadric being 
1669^-57/ + 126t/z + 1001^-452^= 1, 
the directions of the principal axes are to be determined in the 
usual way from the following equations : — 
(1) 3338^-452?/ + 1001 z=rx, 
(2) -452a— 114?/+ 126z=rij, 
(3) 1001^ + 126y = rz; 
in which, since the equations are simultaneously consistent, 
r must be a root of the following cubic 
3338-r -452 1001 
-452 -lU-r 126 =0, 
1001 126 -r 
or 
r- 3 -3224?' 2 - 1602 7 13^ + 52 783878 = 0. 
The roots of this equation, to the first place of decimals, are 
+ 3658-2 +31-0 -465-2; 
each of these roots corresponds to one of the thermic axes 
which are denoted respectively as U", W" , W". As it was 
found that these roots are not sufficiently approximate to give 
the final angles accurately to seconds, the axes were deter- 
