MOMENTS IN GIRDElîS. 79 



By Equations n — 1, n — 2, n — 3, .... r+ 1 



m„ .= • 111,,..,= r^m,, ,^ — — • m,,.,= . . . . ^+ • — ^^ =± — ^ = T 



The coefficients a are given by the same law as for the coefficients a in Case J. 

 Thus, »i,.2 = —-111^. I, and ?)Vfi = "^^ ■ w. 



Substituting these values of m^-a and m,.+i in the r — 1th and rth equations, 

 .'. VI,. X. (4 —^ + m, = — A^m,..,.h-\-mr 



and 7H,.i+wiy. (4 ^^ )^ — B^m,.., + vi,.c 



where 6 = 4——-, and c = 4 ^^ 



Solving the two last equations, 



A.c—B J B.b — A 



.: m^.i^ 1 r smd 7?!^= — -r t 



b.c — 1 6.C — 1 



The ratios -^ and ""'" are each less than unili/, and .'. è and c are each < 4 and >3 



It may now easily be shown that A .c — B and B.b — A are each positive. Hence, 

 m,._i and m^ are both of the same sign. 



The bending moment m^ at any intermediate svipport on the left of r — 1 is given by 

 »), = H — ' • m,.i if q and r are the one even and the other odd, 



or ni, = • m, , if a and r are both even or both odd. 



a... J 



Thus the bending moment at the 7th support is increased in the former case and 

 diminished in the latter. 



If q is on the right of r, 



m, = + "'"-"'^^ . TOr if Ç and r are both even or both odd, 



On-r+l 



or m, = — ^" "''''"' -m, if 9 and r are the one even and the other odd. 



and the bending-moment on the qth. snpport is increased in the former case and dimin- 

 ished in the latter. 



