THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 147 
5. Transformation of the definite integrals into series by means of Cauchy's 
Theorem. 
The integral ¢, of (16) can be written as the sum of three integrals, namely— 
3 cosh kh sinh Kz 32 sf 2 9 *) 
= = + oe S Oa 
$0 I Lys { k(sinh2kh—2kh) 4«*h3 ap k\8h3 i 40 h | 
32 cs = i 27R2,—Kh dk 
Sl, (Jue = 1+ PR) 
i 9 a 5 = ~en OK | 
(at an)|, eR, 
It should be observed that the first and third of these integrals cease to converge when 
R=0. Hence the transformation does not apply to points on the line v=2’, y=y/’, the 
normal to the plate through the sources. 
Consider now the first integral in (18). The function of « multiplying J,«R within 
the integral sign is an odd function of « vanishing for «=0. Hence, as in (2), the 
integral is equivalent to the complex integral 
iN h «hk sinh xz 32 We 8) eG 
= Gye { ae : = | t 
i ae Denno, feu Gm 407) fo” 
the path of integration running from west to east along the whole of the real axis, 
and just avoiding the origin, which is a singular point of G)«R, on the north or upper 
side. | 
On this path take points E, W at distances n7z/2h to the right and left of the 
origin, and on EK W as side describe a square HE W AB in the upper part of the plane. 
The integral over each of the sides WA, AB, BE is easily proved to have zero for 
limit when » tends to infinity through positive integral values. 
Hence, by Cauchy's fundamental theorem, the integral over the path WE is equal 
to the sum of the residues of the integrand at its poles in the upper half of the « plane, 
multiplied by 277, that is, to the series 
oy Gen) cosh «kh sinh xz 
ee xh(cosh 2«h — 1)’ 
the summation extending over the zeroes of the function sinh 2«h —2kh in the upper 
half of the « plane, in the order of their moduli. 
If ¢, is a zero of the function sinh ¢—(, the corresponding zero of sinh 2«h —2«h is 
Kk, =Cn/2h, and 
tm itaR 
Gh =G, == rR eae approximately, 
and we see that this part of ), with its successive derivatives, is practically insensible 
when R is a very moderate multiple of 2h. (Cf. §7, infra.) 
