THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 159 
equivalent for the series (i), and continue the process as far as the continuity of the 
derivatives of f(a, y) will permit. We should thus obtain, in place of ., 
. F 1 cosh kh sinh xz ‘ et ee ee 
(i) (<2. — ame cifcoh Sah 1) | Com VM v delay, 
a function of order h*"*”. 
(ii) A series of line-integrals which we need not write down, corresponding to a 
local perturbation at the edge of A, and giving the edge values of the relative part of 
up to terms in h™. 
(iii) A series of x terms 
Ir(oof— OV f+ eyif--- +++ (CS) CW a 
c ee Syn ; : : if cosh kh sinh xz 
where ¢,, is the coetticient of x” in the ascending power expansion of (— ) AGmiog soe)’ 
and is obviously a rational integral function of z and h of degree 2r+2. If the function 
F(z, y) has its derivatives of every finite order continuous throughout the area A, the 
process can be carried to as high a value of as we please, and we can thus obtain the 
values of p,, 9, to any required order nh. It should be noticed, however, that the 
series (iii) is not necessarily convergent when continued to infinity, as we may see by 
taking as an example f= cos a”, when the series would become 27 cos aa(¢y + cya” + ca*+---), 
which is the expansion (without the terms of negative degree) of 
ates cosh ah sinh az 
a*(sinh 2ah — 2ah)’ 
| and is therefore divergent if |2ah|>|¢,|, G bemg the complex root of smh¢—¢=0 
with smallest modulus. The form of the condition suggests that in ordinary cases the 
series will be convergent if / is small enough; and when this is so, this part of ,, 9% 
| taken along with ¢?,, 9, will define an exact particular solution within A, giving the 
proper values of the surface tractions, and arranged in terms of ascending order in h. 
As a special case, the series will terminate if, for some finite value of n, y”f=0, 
and in particular if f be a rational integral function of «,y. (lt may be noted here 
|that the solution for t=p"t” cosmo might be obtained from the solution for 
FH=Imbp cos me by expanding in powers of 8, and equating coetticients of "+ in 
conditions and solution. ) 
Looking back now to the ¢,, 0, part of the solution, and having regard to (18), 
| (29), (35), we see that we may write symbolically 
B= yf; VE =27v' 7, and 
d, = Ia(cay f-Cav Ss). 
The particular solution to any order in / is then given by 
& = r(euv F—CoV Ftof-Qvrt+----) 
cosh kh sinh xz 
«(sinh 2«h — 2«h) ’ 
well as positive values of 7; or, as we may put it, this particular part of ¢ is given by 
cosh kh sinh xz BIS 5 “5 5 
sinh 2ch 2h)? WTiting — Vv for «*, and operating on f(x, y). 
where c., is the coefticient of x” in the expansion of (-) for negative as 
expanding (- 27) 
