THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 189 
The coetticient is 
3 < 9- 7 2 2 
32h? — 22,0, X, + 22,y,Y, + (a? —y)Z, \ 
II. (v) with m=2 is the above with sin 2,—cos 2 for cos 2, sin 2 and 
coefficient 
3 ¢ 9, gt 
3x ale | — 22, YX, — 22,0, Y, + 22,y,Z, 
miei). Flor! 
( ) oh 
u= —(a+1)zp 
aon Se =(a+1)~| 
(ean log 5 po = pa=0. | 
Coefficient is 
ag eee 
} - ne ane ee 2 
{ ~daaX tn¥y) +( i Rae a 
r ie) | 5 /S2muh?. 
For all the remaining solutions, the stresses are of the third or higher order in 1/p. 
The results of this and the preceding article bear directly upon a principle of 
fundamental importance in theories of approximation, generaily referred to as the 
principle of the elastic equivalence of statically equipollent systems of load, and 
a study of these results will be found of service in imparting precision and 
definiteness to one’s view of the principle in its application to the theory of plates. 
It may be noted here, with reference to the occurrence of the function log (p/2h) 
in some of the principal solutions of § 30, that it would make no essential difference 
if this function were replaced throughout by log (p/c), ¢ being any length whatever, 
the unit of leneth for example. ‘The change would be equivalent to adding a solution 
of the permanent type, giving no body force or traction on the faces, and it will 
be observed that the addition would disappear altogether when the applied forces 
are in equilibrium. 
We have here, in fact, an instance of the mdeterminateness that of necessity 
arises in the absence of conditions at infinity, and we are thus brought to the 
question, what is the exact extent of this indeterminateness? or, as it may be put, 
given one solution of a problem satisfying the conditions at a finite distance, what 
is the most general solution satisfying such conditions ? 
For the investigation of this question we have at hand a powerful instrument 
in Betti’s Theorem, which occupies in the theory of elastic solids the place held 
by Green’s Theorem in the Theory of the Potential. 
32. Bettis reciprocal theorem. Verification of preceding solutions. 
Bettis Theorem may be thus stated:—Given two sets cf displacements of an 
‘elastic solid, with the two corresponding sets of forces maintaining these displacements 
(including body forces, surface tractions, and kinetic reactions), then the work done 
by the forces of the first set acting on the displacements of the second set is 
