454 Royal Society : — 



adding these, and writing t for T + T, which is known, as it is the 

 tangent of the small angle which the neutral lines of the two spans 

 would make at the point 1 if relieved from all load, 



^(A + A'^ + B^ + B'^ + C + C, 

 which may be written 



similarly for the other bearing-points in succession, 



where the number of equations is two less than that of the quantities 

 O , cp 1} &c, so that if two of these are known the rest may be deter- 

 mined. But the first and last are always known, being usually each 

 = 0. Therefore they may all be determined. 



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

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

 the deflection may be 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.=:f i equation (25) reduces 



T =i(^(0); 



similarly, r= ^(^ ). 



I, 

 r 



EI#=F 1 (0 + z(F/), writing i for 

 _i_ l a. l M A ? 



'1+^02+^*0-777^—777^ 



( l i il \^ _i_ l * \iK F il ' 

 -ls + 3> + B* + -S*«-24' , -24' 



Clearing of fractions and transposing, 



8(?+^> 1 + 4^ 2 + 427'9 = / 3 i u+^V + 24EI ^ • • • ( 26 ) 

 an equation which was given by the author in his paper before re- 

 ferred 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 ts=l and t = 0, which is the case of a straight beam of uniform 

 section throughout, 



8(/+/')0 1 + 4^ 2 +4Z> o = Z'V + r/, .... (27) 

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

 moments. 



If in equation (25) we put I— a, it becomes 



T=0e,(^,+ ^ 2 - -!«>,); .... (28) 

 and for the central deflection equation (19) becomes 



Y =^.(-^(^ + f,)+3|- 4 «V,} • -(29) 



