450 E. S. Dana—Association of crystals of Quartz and Calcite. 
The following angles were measured in these “ zones,” as 
they may for convenience be called. A double-telescope goni- 
ometer was employed for the purpose, though the character of 
the planes was not such as to allow of the highest degree of 
accuracy. 
1. Zone i, -R, —R’, 7’ 
t A-~R=113° 12’ ¢ ,~-—R’'=148° 48! -R,A-Rk'= 149’ 28! 
#A-f'=118° 10". i?’ 7 —-R —143° 45' tiat= 77° 39° 
2. Zone i, R, R’, i”, —R", — Rv : 
tAi'= 102° 8’ Bate a 1: 8 eg ee 149° 13° 
Supposing for the moment that the two series of planes 
mentioned are true zones, and also that, as appears at first sight 
to be the case, the whole erystal is a complex twin, it is ob- 
vious that the twinning-plane for the two upper crystals of 
quartz must be a plane in the zone 1, —R, ete., and must either 
halve the angle 747’, or which is the same thing —RA- y, OF 
else be normal to this bisecting plane. Similarly, for either 
crystal above and the one diagonally below, the twinning-plane 
must also lie in the zone 7, +2, etc, and must be either that 
plane which bisects the angle 7a7z”’, and RAR”, or one at right. 
angles to it. 
For the first case the measured angle of —R~—-R’=149° 28, 
gives 88° 24’ that is, 141° 36’ or the angle between the 
twinning-plane and the prism ¢ Again, the measured angle 
77° 29 for iAv' gives 88° 444, that is, 141° 154’ for the inclina- 
tion of the twinning-plane upon the prism 7 : 
For the second case, the angle of R~ R’=124° 2! gives 
128° 58’ for the angle between the prism and the twinning-plane, 
or 141° 7’ if the twinning-plane is normal to the com position- 
plane. Again the angle of ¢,7”=102° 8’ gives for the same 
angle 128° 56’, or, on the other supposition, 141° 4’. If we 
compare this angle (141°-142°) thus obtained with the inclina- 
tion of the prism upon the successive planes of the zone be- 
tween 7 and & having the general symboi m- we are sur” 
m— 
prised to find that it agrees quite closely with that which is re- 
quired for 7. 2-2, viz. 142° 2’. This plane 2-2, one of the most 
commonly oceurring of all the various forms of quartz 1s thus 
approwimately the twinning-plane. In the first case we. og 
tained 141° 36’ and 141° 154, and in the second case 141 Ag 
and 141° 4". There is here some discrepancy, but considering 
4 relation of the crystals the correspondence is very remark- 
able. 
| The conclusion to which we arrive then, is this: that although 
the position of the quartz is unquestionably determined by the 
