178 
-MU. L. X. G. FILOX OX THE EL-\8TIC EQUILIRIHUM OF 
But if be small but still finite, the series 
■la + Ud ^ ^ — g 2c ) COS U 
Se~ 
cos '2n + \u = 
1 + g^S) _ 2e2(i0cos2(6 
As d approaches the limit 0, this series also approaches the limit 0. Hence, 
d fortiori, this series multiplied by d approaches the limit 0, and the stress is con¬ 
tinuous. 
^-d 2c 
Tliis holds provided u fi: 0. But if u — 0 the series in question = --—. The 
when c/ = 0 is — But when d = absolute zero In the series 
limit of 
/■■nd 2c 
Ic 1 
oTid it: 
d2e ^ cos 2r<-|-l?t the series = 0 identically. 
We have therefore in these cases a finite discontinuity in the stress. 
This takes 
|)lace at the points u = 0, i.e., : = fiz ^ ± c, where the shear r: varies discontinuou.sly. 
At all other points rr apj^roaches the value zero continuously as we move up to the 
outer surface of the cylinder. 
Comino; now to tlie distortion of the cross-sections, this is exhibited in Diao-ram 2. 
The displacements are exaggerated, as in Diagram 1, Wq being taken = 4c. The 
cross-sections l.)ecome hollowed out in the middle, the greatest longitudinal extension 
taking place at the sides. Another noticeable feature is that the cross-sections are 
slightly curled round the rim, except over the part of the cylinder v'hich is subjected 
to sliear, where they sloj^e up sharply. 
This follows from the fact that 
iho tlu 
df (7r 
d ' 
Thus, where rz = 0 and dujdz > 0 from Diagram 1, it follows that dwjdr < 0 or, 
since dir/dr > 0 nearer the centre, a maximum value of iv occurs at a comparatively 
small distance inside the “ outer skin” of the cvlinder. When, however, rz increases 
by S, we liave seen that dujdz increases by — S/g, hence dwjdr increases by ^S/g, 
and is always positive at tlie outer surface. At the further end, where S ceases to 
act, the reverse takes place. 
It is now easy to understand wliy the tension is infinite at the inner end of the 
shear ring and the pressure infinite at the outer. For if we take two 2 :)arallel near 
cross-sections, the one just inside the shear ring and the other just outside, the dis¬ 
torted cross-sections remain sensibly parallel until we approach the outer surface, 
wlien they diverge sharply, if near the insitle boundary, and converge sharply if at 
the outside boundary. In tlie one case we get an infinite e-xtension, in the other an 
infinite comjiression. Hence we sliould expect the stresses tc and to become 
infinite at these points, and the stress rr to vary infinitely rapidly—and this, we 
have seen, is what does actually occur. 
Further, we see tliat if we measure the elongation of the outer skin as is done 
with an extensometer, we shall always get too high a value for the extension. 
Beferring to the table on ]). 172, we liave the following table of the di.splacements 
