Mr. J, M. Heppel on the Theory of Continuous Beams. 65 



<f> Qi j v &c, so that if two of these are known the rest may be determined. 

 But the first and last are always known, being usually each =0. There- 

 fore they may all be determined. 



This being so, the bending moment at any point (oo . y) may be found 

 from equations (8), (9), (10) and others of the same form ; and the deflection 

 maybe found from equations (19), (21), (23), and others of the same form, 

 regard being had to the interval of the beam in which the point under 

 examination lies. 



If, now, we suppose that a=b=c=&c. = l, equation (25) reduces to 



similarly, 



T '=w( F ^')); 



(26) 



EK=F 1 ©+i(F/), writing i for i, 



Clearing of fractions and transposing, 



8(Hr tT)fc + 4ty 2 + 4il'<p = Pfi + it V + 24ER . 

 an equation which was given by the author in his paper before referred to, 

 and which is nearly identical with the general equation of M. Bresse, and 

 allowing for difference of notation precisely so with that of M. Belanger. 



If i=l and * = 0, which is the case of a straight beam of uniform section 

 throughout, 



8(/+r>^+470 2 +4Z> o =Z 3 / z+ r V> W 



which is the equation generally known as the theorem of the three 

 moments. 



If in equation (25) we put l=a, it becomes 



T-e^+^-^aV.); • • • • 

 and for the central deflection equation (19) becomes 



If we put b=2a, l=3a, 

 T = a C /\g 7 ^43 7 7 \0 



fl [27*' + ^- a \2T6^ + T08 ft )) 



(28) 



(29) 



13 

 108 



_i_ / 1 , _l 7 . a / 1 ,1 ,11 



(30) 



TOL. XIX. 



