MR. L. N. G. FILON OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
where 6 , 0 ' are proper fractions. If we take e small enough, the convergency ratio 
tends to ■ r . 
4/> 
The series we are dealing with are therefore absolutely and uniformly convergent 
inside a circle whose centre is the point where the concentrated load is applied and 
whose radius is twice the heiglit of the beam. 
The transformation used in the previous section Avas therefore justifiable for this 
region and the expressions (79) are real arithmetical equivalents of the stresses Pg, Qo, Sj, 
Avhich have to l)e superimposed ujDon Flamant and Boussinesq’s solutions for an 
infinite solid when we take into account the height of the beam. The values of the 
first few coefficients, calculated approximately by quadratures, AA’ere found to be as 
folloAvs : Hq = — '2417, Hj = — 'OSOS, Ho = + ’2271, Hg = + ‘3370. 
§ 19. Transformed Expressions for the Displacements. 
If Ave take the expressions (71) fur U and V, we may treat them exactly as Ave 
treated the expressions for P, Q, S. We then obtain, after some rather lengthy 
reductions, U = Uj + Uo, V = W, Avhere 
Ui=- 
1 W// 
7 / 
fJL, 'IttI) 
'■iiL . ax 
e ^ sin ' - 
, W 1 r 1 . ux , 
au — - -7 - - e sm -- an 
27 r X' P yT. Jo /A h 
1 W?/ . W 1 ,, 
— , sm (b —- (p 
fji Ittv ” 27 r X + /u. ” 
1 wy r 
2 - 771 ) Jo 
A 
e '' cos du — 
I 1> 
1 y' ,, , W , 1 
yu 2-77 r ^ 277 \\ + y. 
C ux 
w / j_1- — n 
277 \X' + yU, fj. j Jo '!■<' 
>(83). 
I 
r 
fh 
B, 
du -p Bj 
.^00 I 
Uo = 
1 2XV 
^ 1 / 
+ 'y +i-\- ^ 
r 
- 77 !) Jo!_ siiih^ 2w — 4?d 
'UtC 1 Ulf 
-r cosh ^ 
h u 
3j: 
du 
u~ 
_i_ r__ 
U 77 h ]q siiih” 2 u 
. ux . , uy , 
—7-7 sm — sinh -y du 
■ iu- 1) h 
■y +AU’.__s 
\X' + yu, P / •' 0 sill Id 
2W 1 r 
~ ' ■■ Jo 
77 
77 
+ u 
w + 4 P ^uD 
siiild 2u — 
. rx . ail' X 
— sm — cosh f- — -pQ - 7 - 
2u — All? h h u-h 
. ux . uy' xf 
sm — sinh - — 7% ~ 
h h u-h 
du 
du 
( 83 ). 
