THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 173 
22. The general problem of internal normal force. Approximate forms 
of displacements and stresses, 
The developments given in §§ 14, 15 may obviously be applied in the present case 
also. 
Thus if in (61) we divide by «J,«R, expand in ascending powers of «, put — V’, that 
is — ( S + a for x’, and operate on Z (x, y, 2’), we obtain a form of solution which, 
with the interpretation of V ~‘Z and V~°*Z given in (38), is simply the foregoing general 
solution for areal force of intensity 2u(a+1)Z, arranged in terms of ascending order in 
h?. The solution in this form fails if at (x, y,z), Z or any of its successive derivatives 
become discontinuous, but it has been shown in § 14 how the local perturbation in the 
neighbourhood of any surface of discontinuity may be calculated. 
For the case when Z vanishes outside an area A, the principal part of the perturba- 
tion at the edge of A, when / is small, is found by substituting for Go«R in (63), (64), 
“ = | \ LZ’, y') 2 Gane G(R) = Lig i Ae 
where differentiations and integrations have reference to the accented coordinates. 
Since the solution for the case when there are any finite number of surfaces at which 
Z or its derivatives become discontinuous can be found from this elementary case by 
simple summation, we see that discontinuity in the force itself gives rise to values of 9, 0 
in the perturbation terms of order h’ at the surface, discontinuity in 2 to terms of order 
h? if Z itself is continuous. The next term is of order h* and depends on discontinuity 
of V7Z, that is, of the second derivatives of Z, and so on. 
The symbolical solution for Z force distributed on the plane z=z with intensity 
Qp(a+1)Z(a, y, 2’) per unit area at (x, y, 2’) is given by 
d= + i cosh x(z — 2’) 
sinh xz ee aot a reeee ace ee 
SF (sinh i 2x) («2 smh KZ — 9a + COSN 4K COSN Kz ) 
cosh Kz aa 
«(sinh Qh + 2Kh 
ae (a cosh xz’ + 4 cosh 2xh — a sinh Ki) 
, 
a / Rese o / 
a 92 cosh K(z- 2) + 7 sinh KZ -2’) 
sinh xh 
+ 2(sinh Qich — 2xh 
r c a 
— xz’ cosh 2«h sinh xz’ + = cosh 2kh +44 2x2h? cosh xz’ 
9 2 
cosh Kz , a 5 : 
= - Z . ot! de s 9 9 27,2 7! 
+ (2 (Ginlb 2h + Deh) («2 cosh 2«h cosh xz’ + 4 5 cosh 2«h + 2x«*h? sinh K ) 
with x2= — y?, operating on Z(«, y, 2’) : : ; malls 
The approximate solution is obtained by expanding in ascending powers of «°. By 
retaining only the terms of negative degree in «’, each of the displacements will be 
