166 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
the second order satisfying 
py + (+p) =0 5; py?V ++ =0: py?W + (+ W2=0; 
where 
ee eee 
dx dy di 
(iii) zz=2zy =z =0 onz=th 
It is clear that these conditions do not completely define the solution, seeing that 
no condition to be satisfied at infinity is mentioned. But instead of laying down any 
such condition at infinity, it is simpler in the first instance to be content with any 
solution fulfilling (i), (ii), and (iii). The most general solution can then be obtained 
without difficulty, and with this before us, conditions at infinity can be discussed to 
much greater advantage than at present. 
The problem is solved when U, V, W are found in the ae (4), so as to give 
the same tractions on z=+/h as hese due to (6), but reversed. These reversed 
tractions, as follows very readily from (5), (8), are given by 
wd 
Qu dul — 1 
Tod 2) il 5 a ee C7 pe 
a. 4| 4) dz 
Qn dy \ 
a l+adr™ rt 
Qy Fae eae 
Now when z>7, , 
sll e-k2-2) J cK Rdk 
0 
a -| (= x)e~"@-2) JK Rd 
a 0 
Lae | “ene -OT Rae 
/ 
but when z<2z’, F: 
pa -| ex@-2) J ne Rdk 
=I co 
as | xore-2)J ex 
Uz 0 
Cr es 
re = ks KekZ-2)J 1K Rd 5 : e : ‘ (56) 
Hence if U, V, W be defined as in (4), the function ~ is not required, and the 
conditions to be satisfied by 6, ¢ are, if in the first instance we take integrands 
instead of integrals, 
_~ = «(h—2) ; e-Kh-2)J KR, on 2=h 
5 t — K(h+2') ; e-Kh+2)J cR, on z= —h 
PO dd Bad +a 
FP aa t 3 
— 
K+K(h—2) } e-Kh-2)J cR, on z=h 
alee 
2 
ll ll 
ae ee ee Sees 
| =— 
«-W(dte) beont+O IR, onz= —h . » (57) 
