162 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
a4 ya JyxR - ea it is clear that these equations will all be 
Since JR = -3 7: “ a 
satisfied if we take 
dw’ de dd’ x 
where 
dy! a 
a a JykR on 2=h . ; : . (44) 
= 0 on z=-A ‘ 
, ’ 24 
dv, d¢' Fey aoe J.«R on z=h 
B as 
da dz dz 2 
= 0 on’ Z2=—h : : . (45) — 
a6’ dd’ dd 
Samer Cry = 0 Oil = SE /0 
From (44), 
2 cosh x(z +h) 
= 4g sinh 2«h Fea 
or, separating the odd and even parts in z, 
, 1 cosh xz 1 sinh xz 
eae «2 sinh Kh Jysto= «2 cosh kh ie 
We also find easily = 
. sinh kh 4 
4h = ~ (inh 2eh — Ih) JokR sinh xz 3 
cosh kh . ; Go : 
SPOR ae by ea 
Au’ 2«h cosh kh -- sinh xh 
Be = “2 (sinh Qkh — 2h) 
2«h sinh ch — cosh kh 
: eeu 
J «R sinh xz 
Treating these expressions as in § 3, we find a solution for an element of X- traction 
at (a’, y’, h) of 87 units given by (43) with 
sinh xz 
Fi Zz 
i= ae —Kh— 
a | ( « cosh ee ee - a 
cs cosh xz 1. eh J ee 
ea { Zane eh t els ae laa R?) \ dx 
sinh xh sinh xz 3z e-Kh 3 1 
pie (5 Yor Ss bY nto a aes 
; «(sinh 2«h — 2xh)' fees 4x Se a eG Te Ps ne) \ aK 
ie cosh kh cosh Kz e—kh iT 
i ae (sinh Deh + 2h) + Fe x ele 5 Re+ +78 ) bd 
f 
\ 
* ¢ (2«h cosh kh — sinh «h) sinh «z 32 e-Kh 3 19 
aay ace Pepe 2 ee +f) ty) 
- ffs : (sinh Dich — Qh) on — aR aaa? Ele ae ?) | de 
+ (2«h sinh Kh — cosh kh)cosh xz 1 <7 ( ae iR ‘2 5) ze 
3 
=" . (49) 
a) 
+ (smh Wh Fonh) an T Bn? 
“ 
These expressions may be transformed by the method of § 5, and a slight inspection 
of the relations between (16), (17) and (20), (21) will enable us to write down the 
