on Change of Tem.pera.ture. 413 
At the first temperature let the crystallographic axes have 
directions OA, OB, OC, and lengths A, B, C respectively ; at 
the second temperature these axes will have taken up new 
positions OA', OB', OC', and new lengths A', B', C. As at 
the second temperature the positions of the crystal-lines relative 
to each other are independent of their absolute position in 
space, let the crystal be rotated from its position at the second 
temperature until the axis OC coincides in direction with OC 
and the axial plane OA'C with the axial plane OAC : OA' 
will not coincide with OA nor OB' with OB. 
It will be convenient first to calculate in terms of the above 
quantities the alterations in magnitude and direction of a cer- 
tain triad of perpendicular lines : one, OZ, coincident with 
OC ; a second, OX, lying in the axial plane OAC ; and the 
third, OY, perpendicular to this plane. 
Denote the axial angles BOC, COA, AOB respectively by 
a, /3, 7, and the angle ACB by C. 
I. Let A^/fi^j A,,//^, A 3/ u. 3 v 3 be the direction-cosines of OA, 
OB, OC relative to the rectangular axes OX, OY, OZ (fig. 9). 
Then 
A x = cos AX= sin (3 j X 2 = cos BX = sin EO cos ACB= sin a, cos 
/xi=cosAY= 
v x = cosAZ= cos/3 
fi 2 = cos BY= sin BC sin ACB = sin a sin C 
v„=cosBZ = cos as 
X 3 =cosCX=0 
j u 3 =cosCY=0 
v 3 = cosCZ=i 
II. Let x, y, z be the coordinates of a point P referred to 
the rectangular axes OX, OY, OZ, and &i,yi,Zi the coordi- 
nates of the same point referred to the oblique axes OA, OB, 
OC. Then 
x = \x\ + x 2 yi + A-3^15 
y=/*i«i+#s#i+AVi> 
z=v l x 1 + v 2 y 1 + v 3 z 1 . 
Substituting the above values of \ 1} fj. 1} v x , &c, we find 
x=.x x sin fi + yi sin « cos C, 
y—yx sin a sin C, 
z = x x cos /S +2/1 cos a + z x . 
III. Find as a particular case of the above the coordinates 
(a,'i y Y z x ) of each of the points X, Y referred to the oblique axes 
OA, OB, OC. 
ForX,ifOX=s, 
w = s, y=0, z = 0; 
