272 Mr. G. Wilson. On a Method of determining the 



n% = 22 *47 by calculation. 



n" = 22*53 „ 

 Mean = 22*50 

 Error = 0*13 per cent. 



In practice it is usually very difficult to obtain the value of tlie 

 j*J ~~ dxr by integration unless I is assumed to be constant, or some 



simple function of x. 



The following method will, however, give the values of N, ri, 

 n". . . . required to any degree of accuracy. 



nc.i. 



Let the diagram APCA (fig. 1), obtaiued by erecting ordinates 

 proportional to the values of mjl at every point in the mean fibre 

 AC, represent the new loading. 



The bending moment at any point B may be found very simply as 

 follows : — 



If PN be any ordinate such that FTX'dx represents the load on any 

 small element dx at rT, then if PN be divided in Q so that QX/PQ 

 = 'NC/'NA, then Q'Ndx represents that part of the reaction at A 

 due to this load PNdx at 1ST. 



Hence by taking a sufficient number of points a curve AQG can 

 be drawn to [represent the reaction at A due to the load APC* 

 Again, the load on AB may be replaced by a load at A having an 

 equal moment about B, in a similar manner, either graphically as 

 in the figure, which explains itself, or by calculation. Thus a second 

 curve, ARB, is obtained. 



Then it is easily seen that the moment at B is the difference of 

 the areas of the diagrams AQCA and ABBA, multiplied by AB, 

 that is — 



ISTj = AB(area AQCBBA). 

 This area may be obtained by a planimeter or by calculation, for 



