BEAM OF RECTANGULAR CROSS-SECTION UNDER, ANY SYSTEM OE LOAD. Ill 
w 
here 
— ^0 ( 1)''5;"* -|- ct-j (— l)^ ^ z"^ “ -j- . . . a.2( 
_ 1, 
- »> 
(— l 2 + 4^ — 1) (v/“ 1.? + 4^ — 5) . . . (— l i — 4 ^ — 3 ) 
_+(- \/- l2 + 4^-l)(-12-4-4^-5) . . . ( ^-iz- 4t — 3) 
(— \y z-^ ^ 4 - . . . — aot_ 
2 
■(y— l 2 4- 4^ - l){^/- IZ 4t -i>) . . . (y- Iz - 4(!-3) j 
v""” 1^+4^ — 1) ( — 124-4J —5) . , . — ^ — iz — 4t -\- 3) j 
II we treat in a precisely similar way the second integral L.^„ we find 
4 (201 dz^ 
TT.v 
= T ^YTi ( ^ 2 t (z) tanh — yo, ( 2 ) sech ) • 
(loming now to the case where r = odd = 2^4- 1, we work out K.j+i and L.,,+i Ijy 
similar method. We consider in this case the product of degree 2^ 4- 1, 
{x 4 - 4^ — 1) (.X 4 - 4^ — 5) . . . {x — U — 3) {x — 
wliich we denote by 
60 *“'+^ 4 - hyX-^ 4 - b2X-^-^ 4 - ... 4 - h.t+i. 
After reductions of the same type as those used for Ko,, we rind 
a 
K2( + i - 
(- ly (F+i 
(2^ + 1) ! dz- 
;2t+l 
( 60 (- 1 )^ 2 -'+^ 4 - b.i- 1 )^- 12 “^-* 4 - . . . 
I / J 4.S + 1 4.s' + 3 
■''"'AAo t(4Y + 1)3 + ,-d ~ (4Y + ,3)3 + 
+ (*.(-l)‘r‘«+6,(-l)--'r-+ . . . 
1 
.9'=^ r 
+ ht+i^) S j 
s'=o L 
-.■3 ~t” rjc-' 
=oL(4«' + 1)3 + 23 ' (4++ 3)3 + + 
4^34 +1 - 
(- 1 )' 
{2t + 1)1 d23«+2 
('a(- 
j^y+1 _ 
r 
^^‘^45.1(4, 
.,4 + 
Y +1)3 + 23 ' (4Y + 3)3 + ,+ 
+ {b,{- l)‘r' 4 - . . . 
I 7 f v'" i 4'i’^ + 1 
+ ^2( + l) ^ 
4Y + 
= 0 L(4Y + 1/ + 23 (4Y + 3)3 + 23 
whence writing 
