BEAM OF EECTANGULAE CEOSS-SECTION UNDEE ANY SYSTEM OF LOAD. 109 
K.,, = 2« a-^ t 
* 0 s'=0 
But 
(20! 
(4«' + l)i(. - g (4s' + 3)«| gQg 
4s' - + 4 = ( 4 ./ + 1) — {U — 3) 
and similarly 
Now let ciq, 
4s' — 4 
= (4s'+ l)-5 
4s' 
= (4s' + 1) - 1 
4s' + 4 
= (4s'+ l) + 3 
4s' + 4t 
= (4s'+ l) + (4(-l), 
4s' — 4^ + 4 
= (4s'+3)-(4<-l) 
4s' - 4 
= (4s' + 3) - 7 
4s' 
= (4s + 3) - 3 
4s' + 4 
= (4s' + 3) + 1 
46-' + 4^ = (45 + 3) + (4^ - 3). 
a.,t be the coefficients in the product of degree 2t 
{x + ( 4 < - 1)} [.r + {U - 5 )} . . . [x - (4t - 3 )}, 
when it is expanded out, so that this product is 
rtoX“‘ + foa:-'"' + . . . + as/. 
Then 
: X 
(4f - 1)} _ (4f - 5)} . . .[rr + (4^ - 3)| 
= a^x-^ — a,x-‘ ^ . 4 - a.,/. 
K.>( may then he written 
's Cfo {{4s' + If _ (4,s' -f 3f 
s' = 0 
+ a, {(4s' + If-' + (4s' + 3f-' 
s=0 
+ T Oo {(4.v' + I)'-*-- c-™' - (4s' + 3f 
0 (W' 
»' = 0 
+ . . 
S = » 
+ s as/_: {(4s' + 1) + (4s' + 3) 6-^^'+-’*“} 
s'=0 
■>'= x> 
+ S as/ {6-“*'+'“ - 
8' = 0 
COS (lu. 
