BEA^r OF EECTANC4ULAR CEOSS-SECTTON UNDER ANY SYSTEM OF LOAD. 135 
2L sill </)' L y' . 
jj _ - 2(h' 
TT r ' tt r - ^ 
_SL- yY-*' sm(2^ + l)j,- 
irh t 't I (2v + 1) ! ^ ~ 
4L ® /r' y^ sin 2v^' 4L?/' ^ Aj.'Y>'+i 
TT?; T v7 / (2z.)! + 7753^ f ( 7/ 
_ V (TT TT N 
7 rlr 1 ^\h V" 2 „+i -tio,.; 
sin(2z^ + l)(f)' 
(2r + i): - 
Q = _ 1 4 sin 2f + it s (4 f H, 
TT r'2 ^ ^ 7r5 7 \ a / (2z0 : 
_ / 7 ' 7’'^' siu72^1_) 4L//' - /7 ^2.' sin 'Ivc^' 
7r/r 7 IA y (2i; + 1) ! + -n-Jr 7 H / (2/0! 
^ _ L cos_(f>' _ L //^ cos 20' _ 4L ^ cos (2// + 1) </>' 
tt r' ' TT r'^ 7rA7'7/ (2// +~l)! V^2"+i “ 
4L// - cos 2vcj>' tt \ 1 v 4 os (2// +!)</)' 
■' l" 2 nTi - 4I2O + ^,4 K ^ ^ YoTrrTAV -^2,, 
7rA2 7\ Ay (2/.)! ^ -‘'+1 —“'-'i ' 7rA2 7\ Ay (2/. + 1)! 
where the H’s are given by equations (80) and are the same as before; and 
D = 
u — ^ i — i/A ■*" e 
sinh® 2u — 4//^ ' 4:to 
1 
cosh 10 
+ 11, 
sinh u 
4/a Jo 4(sinli2//4-2/7 
16/0 4r sinh 2// + 2)i 
du. 
du 
The leading terms in U, V, P, Q, S which precede tlie S’s form what is left of tliis 
solution when h is made infinite. They give therefore the displacements and stresses 
due to a shear acting at an edge of an infinite plate. 
They will be found to agree with the expressions olitained l^y Bottssixesq (‘Conqites 
Piendus,’ vol. 114, pp. 1465-1468) for an infinite solid, the strain being two- 
dimensional ; provided that \ be clianged into 
At the point of loading itself the stresses are infinite and the displacements infinite 
or Indeterminate. 
The series in the expressions (107) are easily seen to liave a radius of con¬ 
vergence 46. 
Ihe series for the shear reveals a very curious plienomenon. The terms due to 
the infinite 
plate may lie written 
IT d'- 
They give therefore a positive sliear 
throughout, and zero shear on tlie axis of y. But wlien the otlier terms are taken 
into account, the sliear at points on the axis of ?/ is 
8 = 
4L 
ttA 
?/ 
[j 
(H, -H„)- yH, + 7|(H,-TL) 
4L 
ttA 
•3638 
?/ 
A 
_p -07^^ 
Of> 
/3 
?/p 
Iv 
