﻿374 Mr. C. Chree on the Relation between the 



were independent of r. This, however, is not the case for 

 either hemispherical or plane caps, and it is open to donbt 

 whether it can be secured by caps of any form. The solu- 

 tion (5) thus breaks down in this respect. In accordance, 

 however, with what has been termed the " equivalence of 

 statically equipollent systems of terminal loading"*, it is pretty 

 generally conceded that if one dimension of a body be rela- 

 tively small, such as the diameter of a long beam, then the 

 precise law of distribution of forces applied over the small 

 dimension is not of great importance so far as concerns the 

 elastic strains and stresses, except at points near the surface 

 where the force is applied. It would thus, I think, be gene- 

 rally admitted that, in the case of a closed tube of length not 

 less than twenty times its external diameter, (5) would apply 

 in the case of uniform pressures with a close approach to 

 accuracy throughout much the greater portion of the volume. 

 At the same time it is uncertain how large are the terminal 

 volumes throughout which (5) is appreciably erroneous, and 

 how great is the effect of the consequent error on the result (3) . 

 This error is really neglected by Dr. Guillaume, for though 

 he starts by admitting the failure of (5) near the ends of the 

 tube he applies it finally to the entire length. 



A more serious objection arises when we pass to the assump- 

 tion that (4) holds for two hemispherical caps connected by a 

 cylindrical tube. Let us lay aside for the moment any pre- 

 conceived ideas as to the shape of a thermometer, and suppose 

 we are really dealing with a cylindrical tube with hemi- 

 spherical ends, whose radii E,, B e are the same as the cylin- 

 der's. In this case the theory of equipollent systems of 

 loading cannot be invoked with the least show of reason, so 

 far as the hemispheres are concerned, unless the thickness 

 B e — B* of material be very small compared to B, ; and even 

 in that extreme shape the error introduced by applying the 

 theory would be most uncertain. There is thus, I think, very 

 little mathematical basis for the assumption that (4) may be 

 applied to two hemispheres connected by a cylindrical tube, 

 unless it should turn out that the solutions (4) and (5) are in 

 very close agreement over the transverse sections where the 

 hemispherical caps pass into the cylindrical tube. This, how- 

 ever, is in general far from the case. We see in fact that the 

 radial displacements u in (4) and in (5) are totally different 

 functions of r, except in the special case when Pi=P e . A 

 like incompatibility will be found between the stress parallel 

 to the axis in the cylinder and the stress it would require to 

 equal in the hemisphere. 



* See Todhunter and Pearson's 'History of Elasticity,' vol. ii. art. 21. 



