274 P. II'. Bridgman — Crystalline Cylinders. 



is that they have tetragonal symmetry. Under these conditions 

 a first approximation to the solution may he shown to be of 

 the form ; 



"r =/,('•) +/,(r) cos 40 +/,(»•) sin 40 ^ 



Wfl = / 4 ( r ) cos 4(9 +/;(r) sin 4 6> J 



where ,/ a , /i , /", , and /' D are infinitesimals of the first order, as 

 is also c. Substitution in the stress equations of equilibrium as 

 before gives ordinary differential equations of the second order 

 which may be solved, giving for the explicit form of the first 

 approximation 



u r — Ar+Br' 1 



cos 40 



+ [ 3(a-ft) Br " 1+6, ' S + Hr " 3 + tr " + J '' 5 ] S ' n 4 * 



L af* + ^ - 3^ + "^" + W L ~ ^ J C0S *' 

 + [ -Br~ *- O' + ^D,- 3 - 4 *^ Pr" s ]sin 40. 



This solution contains ten constants instead of six as in the 

 previous case; these ten constants may be so determined in any 

 special case as to satisfy the boundary conditions. The boundary 

 conditions which may be satisfied by these constants are some- 

 what more general in character than those which have been 

 imposed above, for we are in a position by means of them to 

 solve for the case of different arbitrary hydrostatic pressures 

 simultaneously on the external and internal surfaces. 



The solution for the two cases previously given may be ob- 

 tained from the solution for this case by a specialization of 

 constants. 



Trigonal Crystals. — Six Constant Group. The stress-strain 

 relations in cylindrical coordinates are 



rr = c lt e„ + c„e e6 + e 12 e zz + c 16 cos 30 e^ + c„ sin 30. „ 



00 = c v fi rr + c u e gg + c i3 e zz — c I6 cos 30e 9z — e u sin 30e zr 

 ZZ — «,/,.,.+ c n e g g + f 33 e 22 



0z = c 15 cos 30f,.,. — c 16 cos 30e 0e + c Ai e 9z — e 15 sin 30e, e 

 zr = c, 6 sin 30e rr — c ]S sin 30e^ 4- e^e zr -\- c lb cos 30e,^ 



'""'•= — '-,, sin : ^''.r+ C 16 cos 30e M + - (c u — r.Je^ 





