ym. L. X, (4. FTLOX ox THE ET.ASTTC EQIHTJBETUM OF 
Now, if the torsion be symmetrical on either side of the mid-section, we have 
■V — 0, when e- = 0, therefore 
= 0 . 
But also 
Therefore 
Also 
dv 
(hz — iJL T = 0 at the ends. 
^ ' dz 
COS Icc = 0 or Jcc — 2n-j-l7r/2. 
- dfv\ ^ . 7 f/ 
= l^r ( 7) = sin ]:z . 7’ - -y- 
dr 
= S/xA„ sin Icz . /l'Tj {h'). 
By the well-known property of the l-functlons 
iL _ ^“+1 
d.r I ,v» / “ 
Now suppose that the cylinder is subjected to a certain system of transverse 
surface shears, so that nf ) can be expanded in a Fourier’s series in the form 
/ ^ “ . 2)1+ \7r.: 
{r^)a= t c„ sm-jy— 
'hen 
T)!’ 
fxk k„ lo (oi) = 7 
4 _ Al-A 
' " B («) ■ 
1 Fence 
® . 2/1-1 - Itt: 
7’ = X - T ysin .... 
. . . . (127), 
7 l.Hp) • 2 » + Itt- 
. . . . (1-^8). 
0 h(a) 1(‘ 
. . . . (129), 
where p a have the same meanings as before, viz., hr, ha. 
Tn the case where n/c is very small, or tlie cylinder is very long relatively to its 
diameter, we may obtain a first approximation by retaining only the first terms in the 
expression for the I’s, Proceeding as in § 6, we find 
