Bismuth and Cale-spar in Absolute Measure. ~ 867 
where A, B and C are the quantities we have found, a is the 
cosine of the angle made by the axis of the crystal with the 
axis of the bar, and a’ is the cosine of the angle made by the 
same axis with a horizontal line at right angles to the bar. 
he equation 
O=0 
gives equilibrium at some angle depending on a@ and a’, and if 
either of these is zero the angle can be either = 0 or 42, one 
of which will be stable and the other unstable according as the 
y is para- or dia-magnetic. 
For a diamagnetic crystal like bismuth with the axis at right 
angles to the bar we can pu 
éM=cos O= sin} and a=0, 
and we can write 
O=—}a{ 41k,(Lu+Lu"+ete.) +47[(k,-k,)a +k] [Mu+M’u'+ete.]} 
or for very small values of # we can write in terms 0 
© =—2alp {kL + ((k,-k,) a+k,) M} 
If I is the moment of inertia of the bar and ¢ is the time of a 
Single vibration, we may write 
0= 15 tp. 
Tf we hang up the bar so that a’=0 we have 
I 
k (ie Wess 
2Qal ¢? 
and if we hang it up so that a’=4z we have again 
: 7 i ‘ 
AUPh ES tale 
whence 
— wl 
2" Galt, L+M" 
oe dg 
k= — 5 (Gr t 42) 
where 2 
L=A"—sA, A PH(gtA?, +18 A Ayo — S38 A,, Ay + ab Ay! 
M=—A*, - 6A A, P per (354°, 48 SSA, Ay)P+ 222 -8,,Ay Agi i ote 
L+M= 3A, A, P— (8, At, 446A, Ay) P+ ASA, Aol — Ff AVE 
For a cleavage bar of calc spar we must use the general 
©quation. For equilibrium we have 
K,jAa*+ Ba” — Caa’} +h,{A (1 — a’) + B(I-a”) + Caa’}= 0 
Which gives us the ratio of &, to #, For this experiment it is 
st to hang up the bar so that the axis is in the horizontal 
plane and we should then have | 
