(2) 



184 Mr. M. Merriman on the Flexure 



or, universally, 



, , _, .... A« J ,_ r+2 + Ba jy _ r+1 



when n <r+l, U»=<>&*-°» d^^ + Zifo + li ' 

 also 



M.-,=i^M„ M,_ 2 =d 4 M s , &c., 

 or, 



when ?i > ?•, M„=4_ w+2 M s = e? s _ n+2 — / \o/n +1 , /V ( 3 ) 



From (1), (2), and (3) we may obtain the moments at every 

 support due to a load in the span l r . These once found, it is 

 easy to get the shearing-forces and reactions. Referring to (I'.), 

 we see that the expression for the shear in the span l r at a point 

 infinitely near to the rth support depends upon the moments 

 M r and M r+1 and the quantity P(l — k). By exactly the same 

 reasoning we may show that the shear in the span l r at a point 

 infinitely near to the r + lth support depends upon the difference 

 of the moments M r+1 and M^ and the quantity Yh. Let these 

 quantities be represented by a and b, then 



^ '* >for a single load : 

 b=P*, J 5 



a= I wl r (\-k)dk 9 

 b= 1 wl r Mk y 



for a uniform load whose ends are 

 limited by the abscissae k x l r and # 2 / r ; 



a=b = \wl r for a uniform load over the whole span l r . 



Let the shear in the span l r at a point infinitely near to the 

 rth support be denoted by S r , and the shear infinitely near to 

 the r-f lth support by S' r ; then 



~ _ M r — M r+ i f for the right-hand shear attherth-, 



br ~ V~ ~ +a L support, 



, _ M,. +1 — M r , f for the left-hand shear at the r-f 1th 

 k* - I +b \ support, 



_ M ?i — M n+1 f for the right-hand shear at all supports 

 b * - jf— \ except r, 



q/ _M„ — M w _i J" for the left-hand shear at all supports 

 n-1 ~ 4U L except r + 1 ; 



then the reaction at any support is 



R„=S' )i _ 1 fS„, BrBBS'r-,+8.. &C. -. . (5) 



(4) 



