Crystals requiring neither Measurement nor Calculation. 
of the three edges oa, ob, oc, the equation of the plane 5, will 
be, 
X y . z 
4- 
a n> 
H — 
b p,c 
since that plane cuts the three axes, at distances from o, equal 
to W 5 «, 5,^5 c. In the same manner the equations of the 
four planes, 1, % S, 4, will be respectively, 
y , ^ 
X 
n^b ' c 
=: 1. 
X 
y 
a 
n^b 
Pa ^ 
X 
W 5 a n^b pz c 
a n^b p^c 
To express that the intersection of 1 and 5 is parallel to that! 
of 1 and 2 , it is sufficient to write that their projections upon a 
third plane are parallel ; since those two lines are in the same 
plane 1. By eliminating z between the twO first equations, the 
equation of the projection of the intersection of 1 and 2 upon' 
the plane oab is obtained ; 
.{-1 
Inij^p^a m^py^a) in^p^b n^py^b) 
In the same manner, the equation of the projection of the line 
of intersection of 1, ai\d 5, is, 
. { ^ I S i 
Kmy^p^a m^py^a) ^ ^n^p^b n^p^b} p^ p^ 
These equations being those of parallel lines, the’ ratio of the 
coefficients of x and y in the first, must be equal to the ratio of 
the coefficients of x arid y in the second. This, after reduction, 
gives. 
1 
By changing in this equation n^^p^ into n^ p^^ and 
n^’iPz') into p^^ vi^e shall get the equation expressing 
the parallelism between the line of intersection of 5 and 3 , and 
a 2 
