S80 M. Levy m the Determination (^Secondary Faces in 
Although there are three indices, w, p, for each secondary 
face, thenumberofunknownquantitiesisreally only two, -9 and^- 
Two conditions are therefore sufficient and necessary to deter- 
mine them. These conditions are generally the incidences of 
the face upon two known planes of the crystal ; and when they 
are such, calculation alone can determine the law of decrement. 
But in numerous cases, the observation of the face to be deter- 
mined, having two of its sides parallel to two edges of the crys^ 
tal, whose positions relative to the primitive are known, will be 
. sufficient to resolve the problem, without the assistance of either 
goniometer or trigonometry. Thus, in Fig. 11. which represents 
a portion of a crystal, the indices of the planes 1, S, 3, 4, being 
known, those of the plane 5 may be obtained, from the circum- 
stance of its two sides ed, ef being parallel to the intersection 
of 1 and and that of 3 and 4. Before explaining how thi® 
can be done, it is proper to remark, that these parallelisms are, 
in most cases, easily ascertained by the eye alone, from the nar- 
rowness of the planes bounded by the parallels. The dispro- 
portion of the faces of the crystal often facilitates this kind of 
observation. Besides, when the planes of the crystal are suffi- 
ciently brilliant, the reflecting goniometer will readily decide if 
the parallelism does exist or not, and even discover those that the 
eye could not suspect. For it is obvious, that when the crystal 
is so adjusted, as to give horizontally the reflections of an hori- 
zontal line upon two different planes of the crystal, any third 
plane, upon which the reflection of the same line would still be 
horizontal, must be parallel to the intersection of the two first. 
Hence, when the reflecting goniometer is used, after having ad- 
justed two faces of a crystal, it will be of importance to turn it 
completely round, in order to ascertain if there is any other face 
parallel to the intersection of the two first. 
Now, to resolve the proposed problem : Let be the 
indices of the plane 5, relative to three edges, oa, ob^ oc^ Fig. 10. 
of the primitive; ^ 4 , those of the plane 4, and so on. 
What is to be done, is to find the values of p^^ when 
all the others are known. Let o«, oc, be taken for three 
axes, co-ordinates, and a?, z represent the co-ordinates of any 
point parallel to them. Then a, byC being, as before, the lengths 
