THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 169 
Hence if from ¢ in (61) we subtract H/«’+Ke-/«, and from 0, L/x?+Me-*"/« , the 
resulting expressions, integrated with respect to « from 0 to ©, will define a solution 
of the problem stated at the beginning of this article. 
20. Normal force apphed at a single internal point. Solution in series. 
To the integrals thus obtained we can apply the transformation of $5, but one 
remark should be made. From the synthesis which gave (61), it is sutticiently obvious, 
in view of the forms in (58) and (60), that the expressions of (61), with GycR sub- 
stituted for J,(«R), vanish effectively at infinity in the first quadrant of the « plane ; 
that they similarly vanish in the second quadrant follows at once from the fact that the 
functions of (61) are odd functions of «. 
A glance at the relation between (16), (17) and (20), (21) will again save us the 
necessity of writing down the details. Thus, let the values of H, K, L, M, when Ris 
put equal to zero, be denoted by H), Ky, L,, M). Then the persistent part of the 
transformed solution is given by 
p= See imam mena cl 50) 
The decaying part is given by 
G)«R sinh xz 
® = <1 A(cosh 2xh — 1) 
G)«R sinh xz ; ‘ F 1 ; 
aus S _ xz! cosh 2xh sinh Kz + ( 3 cosh 2Ki+ > + 22? jeosh kz i 2 (63) 
, ct , 1 ‘ 
{ ké sinh Kz — 5 (a + cosh 2xh)cosh xz } 
9 = <a(cosh 2eh — 1) 
Go«R sinh xz 
res fae hk : 7 ! 
= <h(cosh 2xh — 1) | kz sinh xz’ — 3 (a+ cosh 2xh)cosh «z \ (—cosh 2h) 
where « is a zero of sinh 2«h —2«h , with positive imaginary part ; 
| with 
» Gok cosh Ke J 
De 7 Kh(cosh 2Kh + Lyi 
G«R cosh xz 
pe 0 é 
a 2Kh +1) 
Kz cosh Kz + (cosh 2«h — a)sinh xz’ t 
; (64) 
1 A 
i xz cosh Kz + 5 (cosh 2«h — a)sinh xz’ i cosh 2«h 
| where « is a zero of sinh 2xh + 2«h , with positive imaginary part. 
When the values of H,, K,, Ly), M, are obtained from (61), it will be found that 
| (62) may be decomposed as follows :— 
: =- - 50x) 
eel 1 ‘ 
O = C= é yx — any? ) 
- | el Sptae sau 1,2) 
(i1) 6= zx each multiplied by EG = /j2)) 4 ae z h 5 2 
see f ee 3 
(iii) oo = =p 
(iv) ¢ =F 4v’x; 0=+}ay’x, with upper or lower signs, as z is greater 
or less than 2’. 
} : ; . : (65) 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8). 27 
