Bismuth and Cale-spar in Absolute Measure. 365 
where Q, 2 ; “ ete. are zonal spherical harmonies with reference 
to the angle 
X= - 4A, Q,+3A,,,Q,,,7° + 5Ay Qyr* + ete. } 
Y=+ {A, Q', ce A,,, Q',, r” + Ay Q'yr* + ete. sin 0 
from which we have the following : 
=A’, Q'+ 94", nA +25 A*,Q*yr+6A,A,,Q,Q,,7 
+10A, Ay Q, Qyr* + 380A, Ay Q,,, Qyr’ + ete. 
= {A’, ne e : "at tAyQ*y r+ 2A, A, Q',Q' 7 
Ay Q’, Qi, r+ 2A), Ay Q’ - ‘Qyr" + ete. { sin’ 6 
XY=-— {A’ a a, + 32, Q,, Q,1-+5A%y Qr Ver’ + (3Q’,Q,, 
+ Q, Q’,,) AA, r + (5Q’, Qy + Q,Q’y) A, Avrt + (5Q’,, Qy 
+ 3Q,,,Q'y) A, Ayr® + ete.} sind — 
The moment of the force tending to increase @ is 
Pe Sado 
Bae 
whence we may write 
O=~4a{A((k,—k,a*+k,) +B((k, kak) —O(k, —h,)aa’} 
where sal, met ore 
B=- +f teak 
d 
SE a oe g : vf Y dr 
C= Del, Sar Fike xX 
where / is half the length of the bar and p = cos 0. 
A=4isin {A*, QQ) + £A%,,Q,,Q 048M QQ TAA (QQ, 
*Q,Q/,)P+A, Ay(Q’Qv+ Q,Qy)P+H48A,, Av(Q',,Qv+@,,Qe)F3 
B= 4lsin 6 { A? (Q’, Q’ sin? 6 — Q” cos 4) + A’, (Q',,, Q’,,, sin’ 9 
— Q” cos oy tay (Q’,Q’, sin’?@—Q’y cos af +A, A.,( (Q’,Q";, 
+Q’,Q,,)sin’é—2Q’,Q’,, cos) + AAr((Q,Q'r + QQ’) sin’? 
— 2Q’, Q’y cos 6 < +A, Ay ((@., Q’y + Q’,,,.Q'v) sin’ 6 
— 2Q’ ,, Wy cos a) t 
C= +41{ A+, ((Q,Q", + Q?) sin’6-Q, Q,c084) +34’, (Q,, 2, 
+ Q",,) sin*— Q,,, Q’,,c086) = + 5A’ ((r Q's — Q"») sin’ 
—QWQreoss) +-8,A,,((90,0, +39',Q,,+@,@,, +2, Q,,,) sin’ 
