CIECULAE CYLINDEES UNDEE CEETAIN PEACTICAL SYSTEMS OF LOAD. 165 
Using the above values I have calculated the stresses and strains for tlie points 
r = 0 , a/5, 2a/5, 3«/5, and 2 := i ( 0 , c/ 10 , 2 c/ 10 , &c.). These are tabulated on 
pp. 171-173. 
§ 8 . Calculation of the Stresses on the Outer Surface of the Cylinder. 
Along the outer surface r = a, p = a, and we have the following ex})ressions for 
the stresses and 22 , and the displacements a and tv : rz and rr ol course are 
known. 
Consider for exami^le the stresses ( 22 ),•=„ and ((/x^ )f^a • 
They are 
and 
4S ^ 67«1q" ~ Cy + ^)h)li — . 2ii + lire . 2)i + lirb ‘In + \ttx 
TT 0 — U + 7«%'K-^ + 1) 
Sin 
sin 
2c 
cos 
• (- 10 ), 
48 <5 27«ld +4(1 — 7 ) L,L — 2(1 — 7 )al|f . 2/i + Ivrc . 'In, + lirh 'In + Itt: 
= y —— - - -i- sin .til 11 - now 
■V 0 ( 7 ^%“ — (1 + 7 A) C") {2n + 1 ) 
sni 
». cos 
Ic Ic 
(50). 
Now, when a is fairly large (say > 10 ), Iq and may be re^daced by their semi- 
convergent exiiansions ; 
I ^ 1 , 
I ^ Sa ^ 2 ! (8«y ^ 3 ; (8«h ^ ■ 
lu(«) = \/ 
^ . /!“ /, 3 3.0 3.5.21 
1 (“) — V ~ g , ^g^^3 . . . j 
(see Gray and Mathews, ‘ Bessel’s Functions,’ p. 68 ). From which Ave find that 
the coefficients of cos in the expansions of ^ {4*^ ),-.=« Rnd (22 ); =«a 2 jpr 
o: 
imate to the values (remembering a = 2 n + 1 ) : 
47 — 2 1 , (2 — 27) (87 — 1) 1 I . 2;«. + Ittc . 2/1 + lirb 
d--- ..rrwo r sm--X sm 
7 2 / 1+1 
T 
{‘In + D- 
2c 
and 
\ 2)1 + 1 
4 2(1 — 7) 1 1 . 2/1 + Ittc . 2/i + lirb 
d- . 1 sm---sin - 
7 ( 2/1 + 1 )- 
•>r 
