gripped between two knife-edges exactiy opposite each other (fig. iii.). ’’I'lie solution 
is obtained from tlje previous one by cljanging the signs of y, \ and S, find then 
adding the new U, V, ]-*, Q, S to tlie old. 
I do not propose tr» write down fully the solution ; it is easily obtainefl ifi va)-ious 
forms by using the several expansions wbicb have already been given f'or tlje. beam 
under a single concentrated load only. The parts of the stresses and displacements 
which become infinite at the points of loading are of exactly the same form as in the 
previous case. 
Let us, however, consider the stresses. We easily find the following expressions: 
2W \ sink //. — v, cosli n lU' , /'// , \ 
— , . , cos , cosh ; (lu . 
irh Jo .SUih 2,u + h' h h 
H.r 
. Il'lj 
,1 , • 1 o , «'T'h —flu. 
TTh Jo h Slllh -t- Zn h h 
e = -^Tf 
nrh J, 
2 W f'® %y sink v, 
Trh Jo h sink 2« -t- 
2W sink M-J-% cosk w v.x , mi 
. — --— COS --cosh da 
’TTO Jo Sink 2«-t-2i<. 0 h 
+ - 
ivl? Jo sink 2a + 
2W f H cosk a 
a sink« a./' . //// , 
--— COS — sinh da. 
2 a -I- 2 a h h 
. v:i: . vy 
, , . , sm —.smh ,, d/a 
TTh Jo sink 2a -f 2a h h 
V 
'a sink 
, - sin' cosh 
tt//' Jo .Sink 2a + 2a h h 
. . (hi). 
The last written equation shows that S = 0 over the plane y = 0. Further, from 
considerations of symmetry V = 0 over this plane. Hence we may, if we choose, 
leave the lower jjaid of the beam out of account altogether, and consider it as 
replaced by an infinite smooth rigid plane, against which the beam is pressed by a 
single weight, W. It then becomes of considerable interest to find out how this 
weight W distributes itself, after transmission through the beam, over this rigid 
plane. 
'I'he pressure — Q on the plane corre.sponding t<^i y = 0 is given by 
. ^ , 2W r^' sink u -f « co.sk n, v.r 
'd = + ■'V - , . ^ cos , d/a. 
irh Jo .Sink 2'/' -H 2'a h 
( 02 ). 
Tt is easy to show that this pres.sure tends to zero when x is large. 
Integr-ating by pards with regard to u, we liave 
2W 
TT.y; Jo 
d /.sink c -k u co.sfi v' . >/./■ . 
T —wt;— rk— j ~r 
d/a \ sink Za Za j h 
The integr-al on the right-hand .side is obviomsly not infinite, however krge :x may 
be. Hence Q tends to zero as x: tends to infinity. 
p 2 
