400 J.D. Dana on Lettering figures of Crystals. 
Thus the ratio m:n: 1 is sufficiently indicated by the expression 
m:n, and m:1l:r, by m:r; or if we use the long and short 
mark (— aud ~) to distinguish the longer and shorter lateral axes, 
:m: 1 may be indicated by m:n and m:1:rby m:n. If 
the lateral axes are equal (as in dimetric forms) this distinction is 
unnecessary and m:n is a general expression for all planes, the 
third term, a unit, being understood though unexpressed. _Nan- 
mann writes these expressions by placing m before a letter P, and 
n after the letter; thus the ratio a: , corresponds to mPn. 
in his system— 
e OFF 2 corresponds to 1 for PF 
i ee er —— yee ilagliia 9 
te ce! tt 4:1 * 4P 
4:2:1 « 4:2 * AP2 
eek: 1 a wrk oF 
m:3 +] ie we 22.02 
we lace - ese“ ePo 
Or in the trimetric system, in which the lateral axes are unequal, 
4:2:1 corresponds to 4:2 or 4P2 
HR eae — 4:2 « 4P2 
Obit: Ort Beaky Lets a a:ae *aPa 
eel ve ies oie “aPx 
But farther explanations of the system are hardly necessary, 
as they will be found elsewhere. ee 7 
In abbreviating this system of notation for the purpose of let- 
tering crystals, I propose simply to drop the P of Naumanu’s sym- 
bols, which is in fact a useless letter, and to write in place of the 
symbol o (infinity) the letter O. The brevity thus attained 
makes the system available, for lettering even complex figures, 
examples of which are here given. 
Thus in figure 2, the three planes 1, 2, 4, are 1P, 2P, 4P, of 
Nanmatw’s system, eqnivalent to 1:1:1, 2:1:1. 4:1:1—The 
plane 0.18 &P or &:1:1.—The plane o-0.is xPx, or a: x:h 
lhe plane 2-2 Is 2P2 or 2: 2:1.—The plane 3-3 is 3P3, oF 
3:3: 1—The plane t-o is Px orl: l:o.—The plane 4-2 is 
4P2 or 4:2: 1. Remembering that the first figure refers to the 
vertical axis, the zones of the planes are easily read off. The 
vertical series 1, 2, 4, 0, represent forms with an increasing ¥eT- 
tical axis, la, 2a, 4a, aa, a stauding for the normal length of 
the axis in a given species. The planes 2-2. 4-2, 0-2 form another 
zoue, in Which the vertical axis increases trina Qa to da and then 
* The units are omitted in this and other cases as unnecessary. . 
b] 
