80 ]\Ir I^evy on the modes of Notation 
_ n + cos CAD — 2 cos (P, P) cos ^ JCAD __ 
1 
2 
^ 1 A • (P» P) P) 
4 cos I CAD sin ^ ^ 
a a 
(P^ P) 
ra + 2cos2 1 CAD — 1 — 2 cos^ J CAD (1 — 2 siu 
" 4cos|CADsbMcos<M " ’ 
(P P) 
and by observing that S cos J CAD sin - ^ ^ 1 % we shall 
; or w = 2 tan COS 2^1.. (C) 
have simply 
i^n • ^») 
S cos 
(P, P) 
2 
A very simple formulae, which will give the value of the index 
n, when the angle {en : e^) will be known, as well as the incidence 
of the faces of the primitive. Let R be the point where the 
sphere described from the point A mets AG, and O the point 
where it meets AB, in the spherical triangle NRO, we shall have 
cos NRO = cos NO sin RNO sin RON — cos RNO cos RON, 
but 
NBO = RNO = ^^4^, NO =(P,a')— 90°RON=60° 
2 
therefore. 
cos = cos ((P, a') — 90°) sin sin 60° 
COS 
2 
cos 60° ; or, by observing that 
(P P) 
cos ((P, a') — 90°) sin 60° cos > get 
cogfe^. i,ut the 
2 
cos = cos sin - I 
formula (C) gives v 
cos 2^ sin ” cos ; hence 
cos (c„ . c„) = ^ cos — J cos and consequently 
* See Mr Brooke’s' Elementary Introduction to Crystallography, p. 36^. 
