Mr. J. M. Heppel on the Theory of Continuous Beams. 63 

 from b to 2, 



I -g-ftJVH; , (13) 



.*. from 1 to «, 



^=6^(^+0, when x=0, F/(a?) = O a ^ =-T ; 



•••|=e 1 F»-T; (14) 



from a to 5, 



g= £2 F 2 '(*)+C ; (15) 



making x—a in (14) and (15), and transposing, 

 C=e 1 F 1 '(«)-e 2 F/(4)-T; 



.-. g =e 1 F 1 »^ a (f;(*)-F,'(„))-T i (16) 



from b to 2, 



g-vP.'«+Cs .(17) 



making x—b in (16) and (17), and transposing, 



C = e 1 F 1 '(«) + e 2 (F 2 '(6)-F 2 («)) -e ? F' 8 (S) ! 



.-. ^=€ 1 F 1 '(«)+.,(F,'(i)-F;(fl))+.,(F,'(*)-F,'*)-T; . (18) 

 .'. from 1 to a, 



y — e^^—Tx, no constant; for if x = 0, F^=0, y=0 ; . . . (19) 

 from a to b, 



y=e 1 F 1 '(a)^+e 2 (F 2 (x)-F^a)x)-Tx+C; (20) 



making x—a in (19) and (20), and transposing, 



C = e 1 (F 1 a-F;(a)a)+e 2 (F 2 (a)^F' 2 (a)a) ; 



l^e^F^HF^^ . . (21) 



from 5 to 2, 



y = e 1 F;(«>+e 2 (F;(%-F;(«>)+ e3 (F3(^-F' 3 (5)^)-T^ + C; .... (22) 

 making x—b in (21) and (22), and transposing, 



V=e x (F x (a) + F»(*- «)) + e 2 [(F a (5) + F 2 '(5)(tf- b)) - (F a («) + F a '(a)(tf- a))] 



+ e,[F,( a r)-(F,(6) + I , 8 , (*X^-*))]-T* (23) 



From the way in which this last equation is formed, it is evident that if 



