Heating and on Double Refraction resulting therefrom. 175 



under the supposition that the temperature was everywhere 

 the same at the same distance from the centre. A similar 

 analysis applies in the two-dimensional problem, which is of 

 greater interest from the present point of view. We suppose 

 that everything is symmetrical with respect to an axis, taken 

 as axis of c, and that 6 is a function of r, equal to *y(a: 2 + ?/ 2 ), 

 only. The displacements in the directions of z and r will be 

 denoted by w and u ; in the third direction, perpendicular to 

 2 and r, there is supposed to be no displacement. 



We may commence with the strictly tw r o- dimensional case 

 where ic — throughout. This implies a stress R whose 

 magnitude is given by 



B -K£ + 3-r'. • • • • (28) 



in which 



~ + - (29) 



dr r 



represents the dilatation. 



The other principal stresses operative radially and tan- 

 gentially are 



P=(X+3,0*L + xH.- 1 y*, . . . (30) 



Q = \J+(X + 2^— 7 0. • • • (31) 



The equation of equilibrium, analogous to (5), is obtained 

 by considering the stresses operative upon the polar element 

 of area. It is 



^^=Q (32) 



dr 



Substituting from (30), (31), we get 



d?u 1 du u y dd 



dr z r dr r l \+2jjl dr ' 



>o 



that 



*L + l=J*-+« (33) 



dr r \-\- ZfJb 



where a is an arbitrary constant. Integrating a second time 

 we find 



ru=— Vf r 0rdr + i«r» + A • ■ ■ ( U ) 



