'270 Mr. G. Wilson. On a Method of determining the 



Let m be the bending moment at any point in the mean fibre of 

 the beam, due to the given system of loading. 



I = the moment of inertia of the section of the beam upon which 

 m acts. 



E = the modulus of elasticity. 



Let the origin be at A ; the axis of x be AqAl . . . A» ; and the axis 

 of y be perpendicular to that of x and positive downwards. 



Also let the suffixes 0, 1, 2, 3,. . . . refer to the corresponding points 

 A , A l5 A 2 , A 3) . ... so that wi 1 is the bending moment at the first 

 intermediate point of support, due to the given loading. 



Then from the equation 



tfy _ _m 

 dx 2 ~ 1 



for the case of a beam supported at each end, we obtain 

 Ey = -jjy dx 2 + ~ET x, 



where T is the tangent of the angle of inclination to the axis of x, 

 of the tangent to the mean fibre at the origin — 



.1 



.where y x is the deflection at A x which would be produced were the 

 beam only supported at each end and under the action of the given 

 loading. 



Again let m', m", m"', &c, be the bending moments at any point 

 in the mean fibre due to the upward thrust of the reactions R b R 2 , 

 R 3 . . . . at the points A l3 A 2 , A 3 . . . . , and T ', T ", T "' the corre- 

 sponding tangents at the origin. 



Then ~E>y x ' = - jj y <foHET % 



B yi " = -jj^^ + ETA 



hence finally since 



y — y +y" +y" + .... for the points A 1? A 2 , A 3 . . . . , 



we have 



<L, \ / rrh. 



ET Q k-^*^dx* = ( ETo7 2 -jfy ^ 2 ) + (ET '7 2 - Jj^^ + . , 



