CHAP. XH.] METHODS COMBINED. 177 



Thus, for 



a=0 a=^l a=\l a=$'l a=l 



x=-OAl x=-0.38Sl x= -0.375 1 o>= -0.357 1 x= -0.3331 



This will be sufficient to construct the curve in any given 

 case. The inflection vertical moves, therefore, between the 

 narrow limits of x = f I and x = % I, or within -J^th of the span, 

 as the load passes from A to B. 



Inasmuch as all that is needed for the determination of the 

 strains in the various pieces are the reactions at the supports , 

 and (for girder fixed at both ends) the moments at the supports 

 also, and as the formulae for the two cases above are very sim- 

 ple, we may determine these quantities at once by interpolation 

 of the given distance of the weight P in the formulas, and then 

 apply the graphical method for the strains, as illustrated in 

 Art. 114. 



Thus, for a horizontal beam fixed at both ends, we have for 

 the moment at the left support A, 



At the right support B, 



For the reaction at the left, 



R A = ?( 

 For the reaction at the right, 



In the case of a horizontal beam fixed at left end and merely 

 "esting upon the right support, we have 



M A = - (3 a? I - 2 a P - a 8 ), M B = 0, 



a being always the distance of the weight P from the left. 

 These formulae are simple, and easily applied to any case. 



We may also observe that in Figs. 79 and 81 the ordinates to 

 the lines P 5, P c, and P c, P B, from A B, are proportional to 

 the moments (Art. 110). These ordinates to the scale of dis- 



