182 Mr. M. Merriman on the Flexure 



rth span only being loaded. Let s= the number of spans, and 

 hi h> lr> &c. the lengths of the spans, the indices counting from 

 the left ; the index n will refer to any support. A single load in 

 the rth span is called P, and its distance from the rth supports 

 is M r or a. Referring to (IV.) we see that there will be two 

 functions of P and kl r that are of frequent occurrence : the one 

 corresponding to the equation for the support r is denoted by A ; 

 the other, for the support r+1, by B. Then 



A = Vl 2 r (2h-SJc*+k 3 ) \ f . ' , , , . 4 , ^. 



' > tor a single load in the rth span. 



B = P/ r 2 (£-F), J 



The uniformly distributed load in l r is w per unit of length ; 

 then, from what is said under (IV.), we have 



A= j wlr{2Jc—M* + k 3 )dJc, I for a uniform load whose ends 

 *^*i > are distant h x l r and hj r from 



If the uniform load cover the whole span, ^ = and £ 2 =1; 

 then 



A = B = \wll for a uniform load over l r . 

 The equations of moments for girders loaded only in the span 

 l r are, from (IV.), since the moments at the abutments are zero, 



2M 2 (/ i + / 2 )+M 3 / 2 =0, 

 M 2 / 2 + 2M 3 (/ 2 + / 3 ) + M 4 / 3 = 0, 



M r _ 1 / r _ 1 -f2M r (/ r _ 1 -f-/r)+M r+1 / r =A, 

 M r / r + 2M r+1 (/ r +/ r+1 ) + M r+2 /, +1 =B, 



M s _ 2 4_ 2 -f-2M s _ 1 (4_ 2 + /,_ ] ) + M,4-i=0, 



M 5 _ 1 /,_ 1 +2M 5 (/ 5 _ 1 + 4)=0. 



The solution of these equations is best effected by the method of 

 indeterminate multipliers. Let the first equation be multiplied 

 by c 2 , the second by c 3 , &c, the index of the indeterminate num- 

 bers corresponding with that of M in the middle term. Then 

 let all the equations be added and the coefficients of M 2 , M 3 , &c. 

 be combined ; then we have 



M 2 [2c 2 (/ 1 + Q + c 3 l q ] ~f- M 3 [c 2 / 2 + 2c 3 (/ 2 + / 3 ) + c 4 l 3 ] + . . . 



-I- M,. [e,._ it-i + 2c r (/ r _ ! + /,)+ c r+1 l r ] 4- .. . . 



4-M s [c s _ 1 / s _ 1 +2c 5 (/,_ 1 + 4)] = Ac r + Bc r+1 . 

 Now let such relations exist between the multipliers that all the 



