420 Haiiy on the Measuring of the Angles of Crystals. [Junz, 
of’ on’, by subtracting 90° from the known angle, and dou- 
bling the remainder. ‘Thus the angle which one of the faces 
”, 2” makes with g, being 153° 25’, as Mr. Phillips found it, the 
incidence of r on 7 ought to be equal to twice 153° 25’ — 90°; 
that is to say, to 126° 50’; and not to 126° 45’ as indicated by 
the reflecting goniometer. This brings it very near the true 
measure, which is 126° 52’ 12”. 
It remains for me to speak of the variety which I call distique, 
represented in fig. 6. The faces x, x, which characterize it, 
result from a mixed decrement of three ranges in breadth and 
two in height on the lateral faces sa 6, s 6 a (fig. 3), from which 
it follows that they have two mutual different inclinations, the 
greatest of which is that of z on z’. These two inclinations 
being given, that of any of the other faces upon g is necessarily 
deduced from them by geometry alone. 
I shall compare here also the three kinds of result obtained by 
the different methods, making use of the same letters as before. 
Incidence of z on x —T, 159° 6’ 58”; C, 159° 6% 40”; G, 
159° 5’. Diff. with T, 1’ 58”; with C, 1’ 40”. 
Incidence of z on z—T, 118° 19’ 24”; C, 118° 18’ 22”; G, 
118° 10’. Diff. with T, 9’ 24”; with C, 8’ 22”. 
Incidence of x or 2 on g—T, 154° 59’; C, 155° 0’ 51”; G, 
155° 25’. Diff. with T, 26’; with C, 24’ 9”. 
The comparison of these results leads me to a remark which 
does not appear to me indifferent. In those which relate to the 
incidence of z on z’, the difference depending on the data from 
which we set out is reduced to a minute and some seconds, and 
thus the simple law of decrement, which, in my theory, deter- 
mines these faces, is confirmed by the three methods. But this 
law being given, the other results relative either to the incidence 
of z on z, or to that of z on g, become corollaries from the first.* 
So that here, as in a great variety of other cases, the crystallo- 
grapher who has calculated one of these angles connected closely 
with a fundamental result, does not afterwards measure it, except 
_ im order to satisfy himself. He would have no doubt beforehand 
that observation, if exact, would agree with theory. 
Yet the measures of the last two incidences by the reflecting 
* If we denote by r the perpendicular cr on ab (fig. 3), by h, the axis c s of 
the pyramid, and by n the number of ranges subtracted, the ratio between the sine 
and cosine of the angle, which measures half the incidence of z on 2’ (fig. 6) is that of 
n—1 
# 1 
272 (—— }) + toh. ane | and with regard to half the incidence of 
n+l n+1 
‘ 
zon z, the corres ponding ratio is that of ./ 7?(n — 1)? + A?n? toh. Further the 
sine of the angle, the supplement of which measures the incidence of z on g, is to 
the cosine as(n — 1) VW 27? + A? to(n + 1) A. In the present case, n = $3 and 
if we make r = 1/ 20, h = 3, we have the results indicated by T. If we make 
r= 1 102,andhk = ¥V 317, we have those indicated by C. But it is visible that 
the angle deduced from the first ratio, being verified by observation, the measure of 
the angles to which the two other ratios lead is unnecessary. 
vi 
a 
