-T47; (24) 



Mr. J. M. Heppel on the Theory of Continuous Beams. 453 

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



0=e 1 (F 1 «-F 1 '(a)fl)+6 2 (F 2 (a)-F' a («»j 



•^y=« 1 (F 1 (a)+F/(«X*-tf))+e,(F,(»)-(P/a>+F;(aX*-«)))-TflT^ (21) 

 from 6 to 2, 



y=€ 1 F/(a>+e a (F;(6>-F a , (a>)+ €k (F 1 («)--F,(*>»)-T»+Cr . . (22) 

 making #=6 in (21) and (22), and transposing, 



y=e 1 (F 1 (a) + F 1 '(«)(* -«))+e a [(F a (5)+F 2 X6X^-*))-(F 2 («)+F>X^-«))] 



+ e.[F.W-(F.(*) + P, , (*X*-*))]-Tar (23) 



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

 that if there were any number of particular values of x to be con- 

 sidered, as a, b, &c, j, k, I, the corresponding values of ^j being 

 e v e a , &c, e w _!, e n , it might be written 



^ = 



■itoto+F^iiXtf-a)) 



+e a [(F 2 (5) + F a '(5)(^-5J)-(F a (a) + F»(^-«))] 



+ e 3 [(F 3 (c) + F>X»-<0) ~ (F,(6) + F 3 (6)(^-i))] 



+ &c. 



+ *«-i[(F„-i« + F n -i(*X»-*)) - (F w _x(i) + F^OX*^))] 



+ e„[F w (^)-(F M (*) + F' n (A)(a?-A))] ; 



if #=Hn (24), ?/ = ; 

 .\T = 



if e l [F 1 (fl) + F 1 '(fl)(/-a)] 



+ ^[(F 2 (6) + F 2 '(^)(/-&))-(F 2 (a) + F»(/-«))] 



+ e 3 [(F 3 (c) + F,'(cX/-c)) - (F,(ft) + F',( W- *))] 



+ &c. 



+ e n _ 1 [(F n _ 1 (^) + F^X*-*)) - (F„_tO*) + F'n-iO*)^-^))] 



+ e M [F H (0- (F B (*) + F^*XM»]" 



If, now, the formation of the functions F^a), F/(a) &c. be examined, 

 it is evident that this equation may be written 



T=A^+B0 a +C, 

 where A and B are known functions of a, b> c, &c. and e v e.,, c 8 , &c, 

 and C is a known function of the same and p lt /i 2 , /u 3 , &c. 



If the adjacent span to the left be now considered, it is evident 

 that a precisely similar equation may be obtained, which may be 

 written T^A'^ + B'^ + C ; 



(25) 



