CIRCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 159 
§ 6. Consideration of the Approximate Expressions to ivhich the Results of the last 
Section lead, when the Ratio o f Diameter to Length is smcdl. 
If we can treat the diameter of the cylinder as small compared with its length, we 
can obtain a first approximation by following the method of Professor Pochhammer 
(‘ Crelle,’ vol. 81), and expanding Aj, Ao, C in powers of a, which is then a small 
quantity, provided the index n is not too large. ^ If we do this we find 
^ 'h±l 1 hi 
47 — 1 a 47 — 1 ’ 
a,. 
7 
47 — 1 
A 7 
« 47 — 1 ’ 
and, expanding I()(/i'^’) and Ii(/i’r) in powers of r, and dropping all the terms except 
the first (-which is really eipiivalent to a second approximation, since the indices go 
up two at a time), we find 
u 
:2(47 - 1) 
/(c) + S () COS hz 
47 - 2 
a 
(-) 9^21 , 
- 2 f47^- 
using the Fourier expansions (38) and (39) 
('■'■)-« - i-y - Oi f X '27Ta 
r __ 1 
47 — 1 2fi 
Noav \{rz)dzX 2TTa is equal to the total longitudinal pull exerted on the bar 
Jz r=a 
by all the fjrces on one side of the cross-section considered. It represents, in other 
words, the total tension at that cross-section. Denoting it Ijy vaKf where Q is tlie 
mean tension at that cross-section, 
u 
X 
X -f- ‘2fi 
2iJb (3X 4- 2fx) 
-Q.7 
X 
2/a (.IX -f- 
(«), 
which shows that the radial displacement is exactly the same as if the only forces on 
a thin lamina between two cross-sections were an external radial tension {rr)r^„ 
and a uniform teiision Q across the plaixe faces. 
* For the aiuilytieal restrictions iiecessHiy in such a case, see §28. 
