93 
BEAM OF RECTANGULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
P 1 , Qi, Si agree with the exj^ressions found by Flamant (‘ Comptes Pveridus,’ 
vol. 114, pp. 1465-1468) and confirmed by Boussinesq (‘Comptes Ptendus,’ vol. 114,, 
pp. 1510-1516) for the stresses in an infinite solid due to a line of load W per unit 
length, in which case the problem is reduced to two dimensions. They correspond, 
therefore, to the stresses that would he induced in the beam by tlie concentrated 
load if the height '2h were made infinite. 
The stresses Po, Q^,, Sg are regular functions of x and y throughout the beam. 
They nowhere become discontinuous or infinite, and they tend to zero as b is made 
large. They represent the correction that we have to apply to Flamant’s and 
Boussinesq’s result as a consequence of the finite height of the l^eam. 
Boussixesq, in the paper quoted above, has made an attempt to obtain such a 
correction, by finding the stresses given ])y ( 77 ) over the lower edge of the Ijeam, 
superimposing an equal and ojiposite system to annul these, and calculating the 
strains due to this last system as if the top boundary of the heam were removed to 
infinity. This corrective system, as lie calls it, will now introduce extra stresses over 
the toj) of tlie lieam. To get rid of tliese a corrective system of tlie second order is 
superimposed, and we may go on indefinitely in this way. Tlie complexity of the 
expressions increases enormously for each system we add, and, on finding the 
approximation so slowly convergent that the terms of the second order were 
practically as important as those of the first, Boussinesq threw up the method in 
despair, and fell back upon an empirical assumjition, given by Sir George Stokes in 
a supplement to a paper by Car us Wilson (‘Pliil. Mag.,’ Series V.. vol. 32 , 
pp. 500-503), namely, that the stress system introduced liy the finiteness of the 
height of the beam was such as to annul the stresses due to ( 77 ) at the lower 
lioundary, and varied linearly along the vertical, giving zero stress over tiie upper 
houndary. The functions Pg, Qg, So of the present article solve the problem exactly. 
I 17. Expansion in Integral Powers about the Point of Discontinuous Loading. 
In the integrals for Po, Qo, So we may expand the quantities x 1 
' RiuJ h sinlij /; 
series as follows :— 
(i h J 
{■2v)'. 
^ "U' (ur'f+H\\p{2v + 1) f 
Sin r- cosh ; - = 
b h 
{2v + 1) 
ax . u 
cos — cosh , 
b b 
H' _ ! ur'f'' cos 2v(j)' 
N 
o' \ h 
u:v a//' Z / 
COS — smh -y = A 
b h 0 
,'\2v+\ ( 3 Qg 2v-rll' 
h 1 {2vPl)\ 
(78), 
u \ "T J-y ; 
being an integer. Now when these values are substituted in (77) and similar 
