﻿Coefficients of Pressure in Thermometry. 373 



u perpendicular to it given by 



i p«r?-p,r; 



W -"3\ + 2,4 Rj-Rj h 



1 



1 P,R?-P e R* , 1 P;-P e R*R* 



u= , w , a — ; — -= — r+ 



(5) 



If we now suppose (5) to apply throughout the whole length 

 Z of the cylinder, whose internal volume is 7rR, Z. we again 

 find the relation (3) to hold. 



Having proceeded thus far Dr. Guillaume argues as fol- 

 lows *: — " En supposant, ce qui n'est sans doute qu'approxima- 

 tivement vrai, que, dans un reservoir compose d'un cylindre 

 termine par des hemispheres, la partie cylindrique et les 

 calottes se deforment d'une maniere independante, on pourra 

 combiner les formules precedentes de facon a obtenir les 

 valeurs qui conviennent sensiblement au cas d'un reservoir 

 thermometrique. La relation (3), commune aux deux cas, 

 doit encore subsister." 



In order to judge of the force of the concluding argument 

 we must consider what has been actually proved. The solu- 

 tion (4) is quite satisfactory for a uniform isotropic material 

 bounded by two complete, exactly concentric, spherical sur- 

 faces. The solution (5) gives a uniform pressure P f over the 

 inner surface r=R £ , and a uniform pressure P e over the outer 

 surface r=R e of a cylindrical tube of isotropic material; it 

 likewise gives a resultant tension 7r(P,-Rf — P e Rg) uniformly 

 distributed over the area 7r(Rg — R?) of any section perpendi- 

 cular to the axis of the tube. 



Now if we supposed a hollow vessel constructed of the 

 cylindrical tube in question closed at each end by any form 

 of surface, or cap, to be exposed to uniform internal and 

 external pressures P» and P e , the resultant of the pressures 

 on one of the caps is necessarily a force 7r(Pj R^— P e R^) 

 parallel to the axis, and by ordinary statics this must equal 

 the resultant of the tensions over any orthogonal cross section 

 of the tube. With a cap symmetrical about the axis of the 

 cylinder, the tension borne by the wall of the tube will 

 obviously be the same at all points in the same transverse 

 section which are equidistant from the axis. The solution (5) 

 would thus satisfy all the conditions for the cylindrical por- 

 tion of the hollow vessel if the caps were of such a form that 

 the tensions parallel to the axis of the cylinder over the trans- 

 verse sections where the cylindrical wall passes into the caps 

 * Loc cit. p. 102. 



