CIRCULAR CYLINDERS ITNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 219 
It is now easy to see, if we bear in mind that when n and therefore a is large, CI^ 
remains finite, as appears on examination of (121), how it is that the stresses at the 
boundary become infinite. 
For in both zz Q and we have, when r = a, terms of order 1/n, when n is 
large. These are of alternate sign, C containing (— 1)”. But if z = c they 
become all of the same sign, and the series become logarithmically Infinite. 
§ 27. Summary of Results. 
Looking back upon the results obtained, we notice : 
[a.) That the three solutions we have ])een considering successively are only the 
simplest of an infinite series of solutions, which are continually growing more com¬ 
plicated ; for we need not necessarily stop, as lias lieen done, at terms of the fifth 
degree, but might go on to terms of any degree in r and 2 ' and thus construct, as it 
were, solutions of successive orders. We should then have an infinite number of 
free constants, which might be determined by introducing further limitations at the 
plane ends, such as, for instance, restricting n to be zero at every point and not 
merely along the perimeter. 
The analytical complexity of such a complete solution would, however, be very 
great, and would render it cpiite beyond the reach of arithmetical expression, and 
consecpiently valueless for the purposes of the engineer and the physicist. No 
attempt has therefore been made to develop this solution, althougii, as an analytical 
possibility, it appears interesting. 
(6.) That the different solutions all agree in giving the perimeter of the plane 
ends as the locus of the points where the elastic limit Avill first be joassed, one of 
these solutions actually making the stress infinite at this perimeter. 
In the more important solution, however, where continuity and finiteness are pre¬ 
served, the conclusion still holds, and, furtlier, is independent of whatever theory of 
tendency to rupture we adopt, whetlier we suppose it due to maximum stress, to 
maximum stretcli or squeeze, or to maximum .shear or stress-difference. 
(c.) That in tlie numerical example considered, plastic deformation begins to occur 
round the perimeter for a stress between 2/3 and 1/2 of that which is re(pTired to 
cause a cylinder under uniform pressure to pass the elastic limit. 
This is apparently in contradiction witli the results of engineering experience, botli 
Uxwix and Perry stating that Idocks of stone or cement, pressed between millboard, 
whicii hinders the expansion of the ends, show greater strength than the same blocks 
when the ends are allowed to expand. 
The key to this appears to be found in a remark of Uxwix, which Professor Ewixg 
confirms, that the lead sheets do not merely allow the expansion of the block, they 
force it, i.e., lead in its plastic state will expand more than the stone or cement would 
do laterally under a uniform axial pressure. 
2 F 2 
