1C) 2 
ME. L. X. C4. I’lLOX OX THE ELASTIC EQUILIBPJT'M OF 
rr 
4,(f) 
^ - :(1 +r)Ai+(i -r)Aj<ioW + 
COS Jcz 
o >4 
cos h 
2A i_^9C 
= ^ - p ^i(p) — 
'H — V 
2C 
{(2y + 1) A. - (2y - 1) A^} lo(/)) + Y pli{p) 
cos /.” 
rz 
H' 
— V 
20 
(Aj + Ao) Ti (p) + — /dIo (/)) 
■?? = 
sfAJ.W+ 7 pT, 
(JS). 
sill I'z 
0 
'll! — 2 AJ|| (/j) + 7 pTj (p) 
0 
sin 
~T” 
III the above a is the argument of the I-fiinctions, unless the argument is written. 
'JA simplify the expressions we shall take ttci = 2c, so that the length is about three 
times the radius. This makes a = 27? + 1 (n = 0, 1, 2, . . . ). Further, suppose 
c = c/6, Ij = c/2, so that the c}dinder is divided into 5 zones, as shown in fig. 2. 
Fig. 2. 
- -€/- ->j<- C/ - > 
a 
-2C 
S 
The middle one from —c/d to + c/'^) unstressed ; two rings from c/3 to + 2c/3 
and —c/3 to —2 c/3 over which a uniform shear is acting; finally, the outer rings 
2 c/3 to c and —2 c/3 to — c, which are unstressed. Also, in order to simplify 
still more, we shall suppose Poisson’s ratio to have the value 1/4, or y = 2/3. 
It may be objected, it is true, that in many actual materials Poisson’s ratio is 
not 1/4. But. this is not really an objection, because the object of this investigation 
is not so much to find out the absolute values of strains and stresses in any given 
material, as to calculate the alterations in these values as deduced from the hypothesis 
of uniform stress, and tin’s we can best do by taking a value for Poisson’s ratio 
whicli is, on the wliole, Avell within the limits indicated by practical results, and which 
makes the arithmetic somewhat easier. 
If we do this and calculafe the values ot the constants, we find that for the first 
10 terms tliey come to the following values : 
