236 Proceedings of the Boyal Society 
8. The form hhl^ where one of the indices is an arithmetical 
mean between the other two, is a double six-faced pyramid, whose 
base is perpendicular to the optic axis. 
First, let 2A = /r -f Z, oy h~k^l — h. 
The faces may be arranged in the following two groups : — 
hlk 
Ikh 
klh 
hkl 1 
• • (A.) 
~hlk 
Ikh 
Idh 
likt ) 
Ihk 
khl 
Ihk 
khl 
} • • 
■ (B.) 
Let the new indices of the face hlh be afBy. 
a P 6 — , y= — ^ 
2l-h-h _4Ji-2h-h-h h-h 
6 6 ^ 
^ = a. 
Hence the face hlh becomes aa/i, where 
2 • 
Again, let the new indices of the face Ikh be a'^y. 
Then 
Now, 
And 
Then = V=i+^ + ^f 
2 > 6 ’ ^ 3 
k—k k—h 7 
= =h. 
Hence the face Ikh becomes aah. 
Similarly, the new indices of the faces klh^ hkl are 'aah and hah 
respectively. 
Hence the group of faces, (A), becomes 
aahj aahj aah, aah 
aah, aah, aah, aah 
h — k 
where a = — if h-k be odd, each of the indices in (A') must 
be multiplied by two. 
Again, let the new indices of the face Ihk be a'/jy, 
Tlinn ' I ~~k ^h — l~k , l-\-h-\-k 
Then a = — , /5 =— , y = 
^h-k —0 —h. 
