CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OP LOAD. IG9 
where 
w,, = 
(l + 7)«V-7«Ip-2IiIo 1 /l-TVn 1 
7 
a\ (2n P 1)^ 
‘In + Ivre . 2)1 + Ivr?* 
Sin —- sm-- 
2 c 2c 
- (I + 7-^) 
and is of order I/(2u -j- l)h 
It so happens that in iv the term of order 1/(2 + 1)^ is evaluable in finite terms, 
and I have included it. 
It is easy to see that 
A 1 
7 (2 h. + 1 )® 
. (2?t + IIttc . (2??. + l)7ii . 2% + Itt.!' 
sm --—^-sm --Sill- 
2 c 
TT" 
hh 
— — —^ from z = — c to z — — h 
= - 2e?> 
1 
9 
(6 + e + zY I from z = — h ~ e to z= —h-\-e 
TT^eZ 
— from 2 = ~l)-\-etoz — l> — e 
= <j 2eh — ^ {b e — zY from z=h — e to z —h-\-e 
IT 
'^eh 
= 7777 from 2 = /> 4- C to 2 = + C . 
luc- 
The leading series in iv cannot, however, be evaluated so easily. It is seen to 
depend upon the evaluation of the series 
0 (2'^i + 1)" 
= U (cot y) + 
,C 
cosec X clx. 
As series of this kind are frequently turning up in investigations like the present, 
1 siii2;i, + li' „ , 
lor values 
C-^ X ® Sll 
have tabulated below the values of d — dx and also of N — 
JoSiiiic 0 (2/1 + 1)' 
f X ranging from 0 to 7r/2 at interva 
olitained by interpolation when re<piired. 
of X ranging from 0 to 7r/2 at intervals of tt/IO. Intermediate values are then 
Table of T [ r— dx = fix ). 
J 0 sin X J \ / 
X. 
fif- 
X. 
f(f)- 
1 
X. 
X. 
f(f)’ 
o 
•039283 
677/40 
•238572 
1177/40 
•450873 
1677/40 
•690354 
2-/40 
•078648 
777/40 
•279605 
1277/40 
•496043 
1777/40 
•743248 [ 
.377/40 
•118174 
877/40 
•321246 
1377/40 ^ 
•542417 
1877/40 
•798291 ' 
iTTjiO 
•157947 
977/40 
• 363596 
1477/40 
•590147 
1977/40 
•855760 
577/40 
•198050 
IO 77 / 4 O 
•406766 
1577/40 
•639400 
2 O 77/40 
•915963 
Z 
VOL. CXCVIII.—A. 
