BEAM OF RECTANGULAE CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 
95 
Now let u be chosen so large that for all values ol v ^ w, where oj is uunierically 
less than unity, 
(2 + IGrt-v”) — (j-nav ^ 
where e is a small, finite, assigned quantity. 
We then find 
L = « 
fl+i 
2v’‘ do 
0 siuh- 2av — 4-a^o 
1 
" f ) 
rX. ^ .CO 
where U,, lies between 4i;'' dv and - 4G' dc, 
1 + 6 j,.. 
Now 
j" yii 4«i; y 
n ; e 
! /* — iuoj 
= — _ (1 4.^ A. . I 
(4«F + 1 \ ^ 1 : ^ 2! ^ • "T 
(4«.ft))'*\ 
“TfT / 
n ! e 
■“ 4 «w 
(4«y + i 
^JiL_ f 
(4«)« + i \ 
(4awy‘ + i , (4«a)F + 2 _ 
(« + iT; + LT+T + • ■ • t« “ 
-4ac. 
1 
“b ... to CO 
Next 
Thei'efore 
(H + 1)! 
sinh^ 2ar — 4rt'r~ > 
r*" 2o'‘ do 1 '“^ 0 '^-* 
Jo sinli^ 2ff» — 4aV Jo ® (d ^ ^ ^ 
(81). 
CO 
n — 3 
^ cd{n - 3) 
Now, (o being > 1, this tends to zero when n is large. Further, by making n 
sulficiently large, the second teimi in (81) is negligible compared with the first. 
We then find that the most important terms in I„ lie between —J— and 
1 + € 4” 4” ■ 
Hence when n is large we may neglect I„_ 2 , &c., compared wdth l„. 
Now 
Ho,. = l2^ + 2; 
where 
il2,.+ l f2i. + 3 “h 2^2i/ + 2 “h 8^2 k + 1) 
f 2d‘'+i(i-c-4'0 , r 
2.+1 = - ■ .2 ^ du < —-5 du < . 
Jo Kinli 2i4 — 4r/ Jo siiilfi2 h — 4r/ 
rherefore H._,„+i = if we neglect all but the most important teniis. Therefore 
in the limit H„ = H+.,. 
Convergency ratio = L 
/•' X 1 1 
I' + O 
l> J r + 1 
= L 
1 + ^6 y i/ + 3 r' \ + 0'e 
1 \ uTTT) ~ 4J d + de ’ 
1 + e'6 
II —cc 
