226 
ME. L. N. G. FILON ON THE ELASTIC EQUILIBRIUM OF 
For r(f) the series is non-iiniformly convergent for r = a in the neighbourhood of the 
sections ^ ^ {c — e) owing to the series having a finite discontinuity. For j>z the 
approximation certainly fails, for r — a, in the neighbourhood oi' z — — e) for 
00 
part of the expression for (f)Z is the series 1 1/(2 u + 1), which is divergent. 
0 
Hence, it we are in such a case to use our approximations for the stresses, we must 
exclude the sections where the applied stress is discontinuous and their immediate 
neighbourhood from consideration. 
It will be found that similar remarks apply to the example of pull given in § 7, 
and also to tlie example given by Professor Pochhammee in his paper on bending (loc. 
cit.), in which he also deals with discontinuous systems of stress, so that his ap^Droxi- 
mate expressions leave us in the dark as to what does really happen at points of 
support, the cross-sections in the neighbourhood of such points being, for reasons 
analogous to those developed above, excluded from the regions where his approxima¬ 
tions hold. 
Before proceeding to an actual numerical concrete case, we may notice that (pz 
becomes infinite at the points s = d: (c — e) for the causes stated above. Hence any 
discontinuity in a system of transverse shears applied to the surface of a cylinder, or 
any such shear transmitted by a grip applied to a portion of the material projecting 
at a sharjj angle, will produce in the neighbourhood a very great stress across the 
section. It would seem, therefore, that a cylinder treated in this way would be likely 
to experience plastic flow, or to rupture, not in the middle, but near the points where 
it is seized. 
§ 30. A 2 ')proximations on the Boundary when the Cylinder is short. 
When the cylinder is short, so that a becomes rapidly large, we may use the 
method employed in §§ 8, 9, and 19, availing ourselves of the approximation (132). 
We then find ;— 
m 
/iTT" 
TT 
T 
1 — 
Ttt 
HC 
ez (from z = 0 to z = c — e) 
and 
ez — ^ (z — c A- (from 2 ; = 
