CHAP. V.] SUPPLEMENT TO CHAP. XIY. 307 



By series, we have (Art. 20) 



_a 8 + 6*^_(Aa e + 6I)4 

 * T - ~~8A^~ ( 



without reference to K, e a = i h. 



For small central angles, then, for which K may be disregarded, the 

 thrust given above by (66) acts at J h below the crown for a rise of tem- 

 perature of t degrees above the mean. For a decrease of temperature be- 

 low the mean it acts above, M is negative, and the strain in the lower 

 flange tensile. 



Further, we have, by series, the approximate formulae 



j. . (70) 

 ' +6*) _15EI f <(Aa+6I)ft 



2r(a 4 H-45 K ) 

 These are the expressions given in Art. 165 without proof. 



