THE EQUILIBRIUM OF AN JSOTROPIC ELASTIC PLATE. Loh 
also suppose that the system ~,¥v, w is maintained solely by tractions on the cylin- 
drical edge, and the system w’, v’, w’ by such tractions along with the force at (a’, y’, 2’). 
Further, it will be convenient to decompose the latter system, and take u,, v,, w, as due 
to a unit X force, U., v2, WwW, to a unit Y force, and uz, v,, ws; toa unit Z force. The 
corresponding tractions on the edge we will denote by X, Y, Z; Xi, Yi, 4; 
eeeey,, Z,; X,, Y;, Z,. The theorem (96) then gives 
ua’, y', 2) = | [xu + Yu, + Zw, — Xyu — Yyv — Z,w)ds | 
UGny, 2) — | [Xu + Yu, + Zw, — X,u—Y,v—Z,w)d8 > . F a (8a) 
BH (Gan 0 by 2) = [ [xu Yu, + Zw, — Xu — Yu — | 
the integrals being taken over the edge. 
As one application of these forms, we may indicate briefly how they can be used to 
verify the single force solutions already obtained. 
Take, for example, the case of a Z force, and let v3, v3, w; have the values defined in 
(63), (64), (65). Also let the edge be the cylinder R= constant. 
(i) The coefficient of the principal flexural term, in which, with the notation of (94) 
Fo (R), is determined from the condition that the resultant of the stress 2 must 
balance the applied force. 
It is interesting to note that the conditions of equilibrium of applied forces and 
surface tractions may be regarded as special cases of Betti’s Theorem. We have only 
to take for auxiliary systems the rigid body displacements w=0, v=0, w=1; u=y, 
a — a, w— 0, ete. 
(ii) In the third of equations (97) take for u,v, w the values of (94) with F=R’. 
_ Only the two flexural terms of (65) contribute to the surface integral; the contribution 
_ from the particular solution ¢ =G,«R sinh «z, 6= —cosh 2«h* must vanish, as we see 
4 by pushing the edge to infinity. 
This, with the result of (i), gives the coeiticient of the second flexural term of (65). 
| (iii) The principal extensional term is verified by taking 
| 
ua ttl @-2'), o= "SW -1), w=(a—3)z. 
| (av) The coefficient of the particular solution ¢=G,«R sinh «z, @= —cosh 2«h> in 
| (63) is verified by taking for u, v, w the values defined by P=J«Rsinh«z, 
| ; 8= —cosh 2kh'. 
| None of the solutions corresponding to the other roots of sinh 2«h —2«h contribute 
| to the surface integral. In fact, the partial contribution from a root «’ being inde- 
: | pendent of the radius of the cylinder, must vanish identically, since the Bessel Functions 
| supply a factor tending to zero or infinity when R is made infinite, according as «’ is a 
| higher or lower root than x. 
| (v) The coefficient of the particular solution p=G «R cosh xz, 9=cosh 2xh'd, may 
| be verified in the same way. 
