REPORT ON THE PRESSURE ERRORS OP 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 f 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 



11 11 



where = -o- + 7T7- f = ~a w 



3n yk on vk 



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

 /c 



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 3 , t 3 , p, and f are to be found. 



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



n 



AA 



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

 With the surface conditions, 



t l = II when r=a lt 

 t 1= r = a , 



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



