
EQUILIBRIUM OF ELASTIC SOLIDS. 99 
the cylinder; 7 is the difference of the angle of rotation of the two indices in de- 
grees. 
This is the most accurate method for the determination of m independently 
of y, and it seems to answer best with thick cylinders which cannot be used with 
the balance of torsion, as the oscillations are too short, and produce a vibration 
of the whole apparatus. 
Cas III. 
A hollow cylinder exposed to‘normal pressures only. When the pressures 
parallel to the axis, radius, and tangent are substituted for p,, p., and p,, Equa- 
tions (10) become 



oe = za) (o+pt+q) a ae a he OME (18.) 
der (Gr -=) (o+p+q) + —p ees A )8) 
d0(7 6) _ Pe 
d(r6) =(5, Bb 35) (o+p+g) + = 4 eeu >) 
By multiplying Bi (20) by 7, differentiating with respect to 7, and 
comparing this value of ld with that of Equation (19.) 
= (5 —\(F dp dq\_1 dg 
rm 9p 3m tanta) m dar 
The equation of the equilibrium of an element of the solid is obtained by 
considering the forces which act on it in the direction of the radius. By equating 
the forces which press it outwards with those pressing it inwards, we find the 
equation of the equilibrium of the element, 

q—p_dp ) 
uF... Ql) 

By comparing this equation with the last, we find 
He - Lean 71 dp dq) _o 
on alae = 52) (Get 7) 
(Ga~sn) on (s+ Fa) a) — 
Since 0, the longitudinal pressure, is supposed constant, we may assume 

Integrating, 

VOL. XX. PART I, 2D 
