BEAM OF EECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
80 
1 espectively. ihe first sum m the second of these expressions does not contain the 
variables, and therefore may be supposed taken from the constant B. The otlier 
terms, addea to the first term of the last line of (67), will give for the infinite 
part of V 
V„=f||(x-=-w"-). 
Proceeding to deal in the same way with the stresses, we find that to ensure 
finiteness in the limit we must add : 
(o) to tlie third and fourth series in P ; ^ 
^ 0cicll CRS0 , 
a 0 46 ^ + 1 )V" ’ 
{h) to the third and fourth series in Q ; S 
«- 
a 7 46'^ (2/i + l)V~ 
respectively ; the infinite part of P is then 
P 
and — 
ft- 
% 
ft 7 4 /Y( 2 ft+l)V~ 
„ Wft 
--iry 
0 — 4 ^3 
Q and S having no infinite parts. 
If we leave out of account the parts Uq, Vy, Pg, which belong to a couple Wa/2 and 
Avhich can be destroyed by introducing an equal and oj^posite couple, we find that, 
when cc is made infinite, the displacements and stresses tend to the following limiting 
values : 
TT _ 1 'Vy P Sillll . lljl. . «// 
~ ' - ~ Sin — smh : du 
/ft lirh J 0 silili 2ft + 2'ft 
cosh ft. 
h 
XIX 
•7 7 I 1-1 h-sm — cosh 
fx irrh J 0 \siuli 2ft — 2ft h h 
3,r \ 
du 
, cosh ft — — ft sinli u ^ 
^ + fx fX 1 . MX . lUl 
-sill - smh — 
siiili 2ft — 2« ft h h 
b,ry 
-j- jJL 
du 
r 1 • 1 1 
I ^ snia u — — cosli u 
< A- -f yft 
silili 2ft + 277 
1 . MX , in/ J 
— sin - cosh -I- du 
u 0 h 
sinli M uv . uy 
' d 7 I “ . —-- COS — cosh -y du 
fx Zvl) J 0 smh 2?7 + 2m h h 
fx 27rb 
cosh 7t MX Ml/ 0 7/1 , 
y 7 T cos - smh — - Y, y du 
siiili 2ft — 277 h h 4&ft^J 
(71). 
y;-+ — ) silili ft ^-ft cosh ft I 
A- + fX ' fJi J 1 
1 
MX . . au J 
COS--smh; du 
77. h h 
uy 
] cosh ft + — ft silili 77. 
siiih 2 It — 277 
\A,' + y'fxj 77/ + U 
1 VU: , 777/ 
— COS -p cosh ~ 
It h h 
' .. ,.2 8 
/X / u 
+ an ai'bitrary constant B' 
^\x' + /x /x) Irid 
•du 
N 
VOL. CCI.—A. 
