294 Mr. L. Fletcher on the Dilatation of Crystal* 
d (1 1), if they had presented themselves, would have trun- 
cated the edges m ?/< 1? / l x respectively. 
In the original paper* Mitscherlich states that, on a variation 
of temperature amounting to 100° C, the angle between the 
planes m m^ becomes more obtuse by 10' 50", the angle be- 
tween the planes I l Y more obtuse by 8' 25", and the angle 
between the edges d a. more obtuse by V 26 // . 
It is worthy of remark that, from the other results given in 
the same paper, we infer that these variations have not been 
directly observed, but have been calculated from the observed 
differences on the assumption that the changes of angle are 
proportional to the changes of temperature and are independent 
of the absolute temperatures at which the measurements are 
made. Perhaps this may partly account for the difference in 
the results obtained by Mitscherlich and Beckenkamp. 
Mitscherlich gives no absolute measurements and no initial 
temperature. We shall adopt the absolute values as deter- 
mined by Neumannf for an initial temperature of about l8f°C, 
and used by him in his own calculations. 
We then have 
First temperature Variations | _ , . 
[Neumann]. [Mitscherlich]. Second temperature. 
am 34° 19 -5 25 dm' 34 13 35 
dl 18 9 -4 12-5 d'V 18 4 47-5 
ad 52 16 -7 26 a' d! 52 8 34 
Let the zone m I (fig. 7 b) meet the plane of symmetry in 
the pole c (0 1) ; we require to calculate ae. 
In the spherical triangle m b I, knowing the two sides m b. 
b I, and the included angle m b I, or the arc a d, we find that 
at the first temperature mZ = 49°0 / 35"'73, and at the second 
temperature =48° 56' 4"'63 ; also the angle m lb at the first 
temperature =59° 55' 31"% and at the second = 59° 58' 
44 //- 5: hence from the right-angled triangle cdl, knowing dl 
and the angle dl c ( = mlb), we find that dc at the first tem- 
perature = 28° 16' 37 //- 2, and at the second temperature 
= 28° 14' 20"-8. 
By addition we have 
ac ... 80° 32' 37"-2, of d . . . 80° 22' 54"-8. 
(a) Find the pole a isotropic to a. From Prop. II. we 
have 
sin dc sin d'a' _ sin 28° 16' 37"-2 sin 52° 8 / 34" 
$~ sin d'c' sin da ~ sin 28° 14' 20"-8 sin 52° 16' 0"' 
* Abh. d. k. P. Ah. zu Berlin, 1826, p. 212. 
t Pogg. Ann. 1833, vol xxvii. p. m 
