CIECULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OE LOAD. 233 
No doubt the cases treated involve somewhat arl^itrary conditions, not strictly- 
obtained in practice, hut it appeared useful to ascertain how far they gave results 
diverging from those which would be found on the ordinary hypothesis of uniform 
extension or torsion. 
This furnishes us with a test of how far we may accept de Saint-Venant’s 
principle of “ equipollent ” systems of load for a bar whose length is gradually made 
smaller compared Avith its diameter. The results Ave have here obtained indicate 
that, as we go aAAmy from the points of application of the stress, a “ uniform ” solution 
is reached much sooner in the case of torsion than in that of either tension or 
pressure. 
With regard to the arithmetic of the paper, the results have l)een as far as possible 
checked. It is believed that they are correct to the number of figures given, but 
owing to the sIoav convergence of certain of the series, accunudated errors may in 
some cases affect the last and even the second last figure. Ea^oii this, hoAvever, Avould 
not sensibly disturb the coiiclusions. 
For the I-functions the tables in Gray and Matheaa^’s “ Bessel’s Functions” were 
used, but the range of the tables is so limited that a large number of these functions 
had to be independently calculated. The semi-coiiAmrgent expansions were employed, 
the argument being large in each case. 
My very best thanks are due to Professor Ewing for his unfailing kindness in 
coming to my aid Avith suggestions and advice. 
2 U 
VOT.. CXCVIII.— A 
