148 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
As to the other two integrals which occur in (18), we have proved in (h) that 
i (JyeR— 14 jePR%e-1) TE TR? log 2 — TRE, 
R 
and |, onk — em) — tog 
These functions will occur so often that it will be convenient to reserve an invariable 
symbol for the former of them, say 
Sp 
R 
5 oh | 
x(R), or simply x, = i lope | 
and then v2x=log am 
The persistent part of p, is therefore 
3z ee RIN 5 
ajax * & * 40 iY a 
which is the sum of two potential functions, 
2 Ou ie elles *x) 9 
alex ge? Vx and 017 X° 
6. Types of the particular solutions composing the general solution. 
A glance at the relation between the results just obtained and the form of ¢) m 
(16) enables us to write down at once the corresponding transformations of 4, ¢,, 
Collecting the results, we find 
eg Bi cosh kh sinh xz 3 (« 1s 4 9 co 
: «h(cosh xh — 1) ° 4h3 ee Gr aX 40h WX 
ee 2G, RCosh xh + 2kh sinh «h)sinh Kz 
1 3—2 Dh?2: 2. 9 2. (20) | 
xh(cosh 2xh — 1) ih TAA Oo ek x) “Agno * 
where « is a zero of sinh 2xh —2«h, with positive imaginary part. 
oa) pees + hot 
Ligh (cosh 2«h + Spe 
bys = 6. Risin kh + 2xh cosh kh) cosh xz 3_, ; : - (21) 
kh (cosh 2xh + 1) a 
where « is a zero of sinh 2«h + 2«h, with positive imaginary part. 
The solution must give iz, 7, % all equal to zero at the two plane faces of the 
plate, except when R=0, and we are thus prepared to find that the strain defined by 
the terms corresponding to any one root « gives zero stress across z=-+h. Thus in 
(20) >) contains a series of particular solutions of the type 
(i) p= — cosh kh sinh xz F(a, y) where (y?+«?)F=0 
6= (cosh kh + 2«h sinh kh) sinh xz F(a, 7) } sinh 2xh — 2xh =0 \ 
(22) 
