THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. US, 
Here VF, =0, and if we write (4+ 1)F for F,+1h’V”F,, these expressions reduce to 
the form which we have taken throughout for this kind of strain, namely, 
oe ‘ 
U(x, YZ )= ar 
=(a+1)(¢F -22°V?F) + 2(229-1%') VF 
3) er are (101’) 
eee | 
we, y', Z)= (a+1)(F-32?V2F) + 2(2?-22)V °F a 
where V'*F=0. 
(ii) Transitory modes, or rotational type. 
Put F, = work difference from the system = eae s——a, Sinh «zGy«R. 
Th , , rr 2 , 
ea HORI ea 2 a) 
Pees" 2 102 
ec aie 
we, Y,2) = 0 
es E, d?F 9 2 
where « is a pos. imag. root of cosh ch, and 3+ aj * l= 
The solutions are of the type ~ = sinh «z he SY). 
(111) Transitory modes, 6-¢ or dilatational type. 
( bo sinh «zG,)«R 
Put F,= eyotk difference from the system 8rp(a+ 1)K?h(cosh 2«h — 1) 
\ 6 = -cosh 2xh* > 
kK ¢ lap" 
2x2’ cosh Kz’ + (a — cosh 2h) sinh xz’ 
oa, y's 2) 5-3 aE (103) 
w(a, y', 2) = > : { 2x2’ sinh xz’ — (a + cosh 2«h) cosh xz’ 1 
K 
PF, 
where « is a zero of sinh 2«h — 2«h with pos. imag. part, and & aun ay oe HO 
The solutions are of the type = sinh xz’ F,(a’, y’), 9@= —cosh 2h ¢. 
34. Form of the solution for edge tractions deduced by another method. 
We have thus shown that the most general deformation of a finite plate under edge 
tractions only is compounded of the types specified in (98)... . (103). The deforma- 
tion is of the same form as that given by our infinite plate solutions for any part of the 
solid free from body force or surface traction, and it may be of advantage to show in a 
direct manner why this should be so. 
Suppose, then, that we have given a displacement (wu, v, w) of a finite plate bounded 
_by an external edge S and one or more internal edges 8’, the only applied forces being 
tractions on the edges. Imagine the plate continued inwards and outwards so as to 
