THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE, 175 
If we denote — pet 1)F, the principal term in w in (72), by W, we have 
7 
3 
ay a0 
NE Cores: alia 
3 : 
ee eG LINZ 
lege 
3 A+ 2hg 
Buh? A+p 
Cyt*W=Z, where C= $yuh?(A+ p/(A+ 2p) : ; pe ee(Wit) 
or 
In this notation, to a first approximation 
Ae 8G ae a) 
On \ det dy? 
Ay 3C Bal a) 
YY ORE” de® dy? 
Rane 2W 
1 1- 4———- i 
Been) dads iD) 
Acain from (72), 
a de dw Sp a & 
2 - 4(2 — h?)— vy 7I 
=(7 Fels 32rph? a res 
= 70 - i) = yaw ; ; 2a) 
23. Normal force a function of z only. 
It may be useful to put down here the next term in the development of w, of which 
_the two principal terms are given in (72). This is 
a — 3)2 Cit Ors 4 Weel 
Vv F/@ nse +2 14 4% 32y'2 = a a0 = ( 4 a+ ns 
ae at ere oy veneers 0 aaa =5¢ 3.527) 78) 
5 eGo "heae! 
as((l —a)(z-2’) L(x, Y, z) : 
n this, of course, 
VF =27rZ(x, y, 2) 
‘he terms which have to be added to (72), (78) in order to give the complete particular 
| values of u,v, w, all contain x,y derivatives of V‘F or Z. Hence, if Z(a, y, 2’) isa 
| function of z’ alone, (72) and (78) give a complete particular solution of the problem. 
urther, Z may have one constant value in one region of the plane z=2’, and another 
} constant value in another region of that plane. (72), (78) will still give a particular 
| solution in each of those regions taken separately, or rather in the cylindrical spaces 
| of which these regions are sections, but it ought to be carefully noticed that it is not 
| in general an exact solution when the two regions are considered together as part of 
|one body. The point of failure is, it need scarcely be said, the condition of synexis ; 
