77 
not have led to any result that has not been obtained by the 
more usual method of treating the subject, an acquaintance with 
it can hardly fail to impart a clearer insight into the compli- 
cated relations of crystalline forms, and afford a fresh instance of 
the truth of the remark made by Sir John Herschel (Astronomy, 
7th Kdition, p. 6) that it is always of advantage to present any 
given body of knowledge to the mind in as great a variety of 
lights as possible. 
2. According to Grassmann, if from any point within a 
crystal lines be drawn normal to the several faces of the crystal, 
and any three of these normals, not all in one plane, be taken 
for axes, the equations to any other normal will be 
se Nt TENE: 
ne TIE i? 
where a, 8, y are three straight lines the ratios of which depend 
upon the species of the crystal and the selection of axes, and 
h, k,l are any integers either positive or negative or zero, one at 
least remaining finite. That these two enunciations lead to 
identical results, though not at first sight obvious, admits of an 
easy proof. 
3. In fig. 1 let O be any point within a crystal, Let the 
surface of a sphere described round O as a centre meet the 
axesin X, Y, Z Let ABC be the polar triangle of XYZ, and 
therefore OA, OB, OC radii normal to the faces 10 0, 010, 
001; Pthe pole of the face h & I; and a, 6, c the parameters of 
the crystal. 
* Let LZ be the intersection of the great circles BC, AP. 
Through any point R in the straight line OP draw RQ parallel 
to OA, meeting the straight line OL in QY. Through Q draw 
QN parallel to OB meeting OC in N. Let QR=2, M Oy, 
ON =z. It is proved in my Tract on Crystallography (4) that 
jede 
, sin BAP= ‘sin CAP. ’ sin CBP = : sin ABP, 
[Reprinted, 1880.] i 
