154 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
11. The same special problem. Summation of two infinite series. 
An important part of the strain given by $,, 9; remains to be considered, namely, — 
that which arises from the term 27/(8°—«°). J,,8e cos mw in the value of the surface 
integral for the case when the point (p, ) is within the cylinder p=a. Denoting 
these parts of ,, 9, by $,, 9,, we have 
: cosh xh sinh xz 
(2? — «*)kh (cosh 2«h — 1) 
re hey we (cosh xh + 2«/ sinh xh) sinh xz 
Bee ate anne 2 a (B? — )h(cosh 2eh — 1) 
We note in the first place that ¢, admits of three, and 6, of two term by term 
differentiations with respect to z when —)<z<h, while a, y differentiations can be 
performed without restriction. Combining this result with those already obtained, — 
we see that in the complete solution in terms of surface integrals, all the differentiations 
necessary to give the displacements and strains or stresses at any point in the body 
of the plate can be performed on the series term by term. 
When / is small, , and 6, are of order h’, and a 2z-differentiation lowers the 
order by one. ‘This can be seen from the series, or otherwise, for, as we shall now 
show, the value of the series can be found in finite terms. 
Consider the function of x, 
3 = 27J,,Bp cos mw 2 =) 
(31) 
cosh «kh sinh Kz 
(B° — x*)k(sinh 2xh — 2h) 
This function, multiphed by «, vanishes at infinity at all points of the path 
EW AB described in § 5; hence the sum of its residues vanishes. The function being 
odd in «, the residues at the poles «= «’ are equal. Thus 
2 (series of residues at zeroes of sinh 2«i—2«h in upper part of plane) 
+ 2 (residue at «= 8) + (residue at x=0)=0. 
The residue at «=8 is 
cosh Bh sinh Bz 
(=) 28°(sinh 28h — 26h) ” 
Also if : 
coshkhsinhke — <A, B 
Mise) wel ce 
near k=Q, 
then the residue at «=0 is 
sees 
pat Be 
> (-) cosh xh sinh xz cosh Bh sinh Bz at B 
(8? — x’) kh (cosh Deh =I) F ~ BXsinh 2Bh—2Bh) pt + BF 
It may be noted that A/8'+B/6 are simply the terms of negative degree in the 
expansion of 
Hence 
cosh Bh sinh Bz 
B*(sinh 2Bh - 2Bh) 
in ascending powers of £. 
