THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. eat 
In order to sum these last series, consider the function of «, 
a cosh «(z— 2’) . 1j ; 
TaD LON a zZ ear 
+ (a) (+ according as z> or ) 
sinh xz Base ‘ j 
+ Ca CNT) { Kz sinh xz’ — $(a + cosh 2xh) cosh xz’ t 
cosh xz : ; : ; 
ar (B= @)«(sinh 2kh + 2xh) { Kz cosh xz + 4(cosh 2«h — a)sinh xz \ : ; . (68) 
Looking back at (61) we assure ourselves that this function vanishes at infinity in 
such a way as to make the sum of its residues zero. Also, since the function is odd in 
x, the residues at «= + «, are equal, and therefore the sum of the residues at the zeroes 
of sinh 2xh + 2«h is simply the coefficient of 27J,,8e cosme in (67). The sum of the 
residues at «= + @ is 
_ cosh B(z— 2’) 
ren G9 
sinh fz “pee a ah 9 Saran 
BXsinh 2Bh — 2Bh 2Bh — 2Bh) | Bz sinh Bz — }(a + cosh 2Bh)cosh Bz f 
cosh fz P f E 
AXsinh 2 ~ + 2Bh) | Bz' cosh Bz' + (cosh 28h —a)sinh Bz : wen(G9) 
: ’ A B : 
If this last expression near 8 =0 be of the form git = 2, the residuerat <— @ 
of (68) is simply - zt ~ap. 
Hence the coetticient of 27J,,8e cos mw in (67) is simply (69) with sign changed and the 
terms of negative degree in § subtracted. These terms of negative degree, just as in 
§ 12, are added on again when we take in the part of the solution coming from (65), 
which is obtained by writing F for x in (65) where 
F | [x@InBe' cos mw p' dp’ da’. 
2 
ur 
= B JmBp cos mw +F,. (Introd. (/).) 
The term ai nBp cosmw being taken in for the purpose just mentioned, we are left with 
F, instead of x in (65). Since y*F,=0, these equations now define a combination of 
deformations of the persistent or permanent type, under no body force and no surface 
traction. 
The solution therefore resolves itself into 
(i) this free deformation of the permanent mode ; 
(ii) a local perturbation ; 
(iii) a particular solution giving the proper discontinuity of stress correspond- 
ing to the applied areal force. 
