78 BOTEY ON THE BENDING 



Similarly, by Eqs. 1 and 2, 



m,, = — - ■ Wj + ■«'3 + 7 •4.w)i + î«2+ 15.mi 



r^ 



«Li-l 



ja„_4.a„.j+4.M'i-3a„_i.a„_,+ 12.?('2+lla„.i.a„_4+44.M'3+ | 



5, iVg, are greatest 



w, , Wg , :. .. are least. 



Thus, the general principle may be enunciated, that, " in a horizontal continuous 

 girder of « equal spans, with its ends resting upon two abutments, the bending-moment 

 at an intermediate support is greatest when the two spans adjacent to such support, and 

 the alternate spans counting in both directions, carry uniformly distributed loads, 

 the remainder of the spans being unloaded." 



Case II. — The principle deduced in Case I also holds true when the loads are distri- 

 buted in any arbitrary manner. 



Consider the effect of a weight w in the r-th span concentrated at a point distant p 

 from the r — 1-th support. 



By the Theorem of Three Moments, 



4.m,+?n, = ■ (1) 



m.+4.m,H-3m=0 (2) 



m., + 4.m.^+m^^0 (3) 



OT,_2+4.mr_i+m,.. = — W-- -r— y^ — yl, suppose (''—1) 



' WV.1+ 4.m,+ m,^.i = — ?t'- ^^' • {P-l-p'') = —u> • ^T^.2.l-p = — B, suppose (r) 

 TO,+ 4.m,+ i + »v_^.,= ('■+1) 



w«,.-2 + 4.m„_o+??i„_]=0 («— S) 



m„^2+4.m„_i =0 (w-1) 



By Equations 1, 2, 3 .... r — 2 



111 1 m,.i 



4 15 00 a„_2 a^-i 



the upper or lower sign being taken according as r is even or odd. 



