ON THE STRESSES IN SOLID BODIES DUE TO UNEQUAL HEATING^ &C. 39 



The otlier principal stresses ojjerative radially and tangentially are 



p=(;.4-mJ^+V-7^. m 



(Ir r 



The ccpiittion of e(iuilihriuinj an;ilogous to (5)^ is obtained by consi- 

 (h'riiig the stresses operative u])ou the pohir chôment of area. It is 



V-- (-) 



Substituting froui i^M)), (31)^ we get 



d'^io 1 di( u y (M 



r/72 '^~r~(ïr~ r^ ^ A + 2,'^ Jr 



so that 



du . H yû , , ^ 



dr r a4-2,v. ^ ' 



where x is an arbitrary constant. Integrating a second timc we find 



ru= T^-\'0rdr^kocr--{-i3, (34) 



m \vhicli, liowever, /3 inust A'aiiish, if tlie cylinder is complète througli 

 ;. = 0. Trom (3L) 



F = [?. + !..)x-^^U'' ùrdr,. (35) 



and 



1'- ^^ = 74^-!^-4P^^v/rj (37) 



It is on {F — U) tliat the double refraction dépends when light tra- 

 verses the cylinder in a direction parallel to its axis. 



