REPORT ON THE PRESSURE ERRORS OF THE THERMOMETERS. 27 



When this method has to be extended to pressures such as would crush glass, recourse must be 

 had to steel. A number of steel instruments, in their turn, can have their scale units determined 

 accurately from one another, each from a thinner one ; until we come to the thinnest, whose unit is 

 exactly found by comparison with one of the thicker of the glass instruments. We have thus a series 

 of gauges, each of any desired sensitiveness, capable of reading accurately pressures up to those for 

 which steel at the interior of a thick tube ceases to follow Hooke's Law. 



To illustrate this process, and to show what amount of sensitiveness is to be expected from an 

 instrument of known dimensions, I append an approximate solution of the problem of the compression 

 of a cylindrical tube with rounded ends. The exact solution would be very difficult to obtain, and 

 would certainly not repay the trouble of seeking it. I content myself, therefore, with the assumption 

 that all transverse sections are similarly distorted ; which, of course, involves their continuing to be 

 transverse sections. 



Let £ denote the displacement of a transverse section originally distant x from one end, and let 

 p be the change of r the original distance of any point of the section from the axis. Then, as it is 

 obvious that the principal tractions are along a radius, parallel to the axis, and in a direction perpen- 

 dicular to each of these, we have at once 1 



c Jf = 4 -ft, -ft, , P = -ft, + ct 2 -ft, , §■ = -ft, -ft, + et 3 , 

 ar r ax 



where c = t; — |-7rr» /=T TvT- 



3n 9k J bn 9k 



Here -=- is the compressibility, and n the rigidity, of the material of the tube. 



In addition we have for the equilibrium of an element bounded by coaxal cylinders, planes 

 through the axis, and planes perpendicular to it, 



and the approximate assumption above gives 



-^ = constant. 

 ax 



From these five equations t lt t„ t 3 , p, and f are to be found. 



They show that t 3 is constant, and its value must therefore be 



2 



-n- ' 



a,— at. 



where II is the pressure, supposed to be wholly external. 

 With the surface conditions, 



t l = — LT when r = a, , 

 t 1= „ r = a , 



Thomson and Tait, Nat. Phil. §§ 682, 683. 



