190 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
equal to the work done by the forces of the second set acting over the displacements 
of the first. 
In potential theory one of the chief applications of Green’s Theorem is to the case 
when one of the potential systems includes a mass concentrated at a single point, and 
in the present subject Betti’s Theorem finds an application of like importance when 
one of the displacement systems contains a finite force applied at one point, or, in 
analytical language, includes a point singularity of the first order, that is to say, of one 
of the three forms indicated in (6). 
Thus, let us suppose the solid to be bounded by a surface 8, and in the first set let 
the displacements be u, v, w; the components of body force per unit volume X, Y, Z; 
and the components of the traction on 8S, F, G, H; in the second set let the displace- 
ments be wu’, v’, w’; the only internal force a force X’,-Y’, Z/ at (a’, y’, 2’), and the 
tractions on S, F’, G’, H’. . 
We may apply Betti’s Theorem to the space bounded by S and a sphere 8’ of radius 
e drawn round (a’, y’, 2’) as centre. Thus we have 
| | i (Xu! + Yo' + Zw'\dV + | few + Go’ + Hw')dS + | | (Ful + Go’ + Hogs’ 
= | for +G’v+H'w)dS + [fer +Gv+Hw)dS’. 
Now take the limits of both members of this equality for «= 0. 
Since near the centre of the sphere S’, wv’, v’, w’ are of order 1/e, F’, G’, H’ of order 
1/e*, and dS’ of order ¢’, the effect on the volume integral is simply to extend it to the 
whole volume within 8; the surface integral | i (Fu’ + Gv’ + Hw’)dS’ vanishes, and the 
surface integral i i (F’u+Gv+H'w)d’ has the same limit as 
u(x’, y's d)| [ras + v(a’, 7’, dy] [Was’ +m, y, )| pas, 
namely, 
ula’, yy Z)X +(e’, o', ZV +(e, y', 2S, 
the tractions F’, G’, H’ on S’ being statically equivalent to the force X’, Y’, Z’ at its 
centre. 
It is thus apparent, and might indeed have been anticipated, that Betti’s Theorem 
may legitimately be applied when one of the systems contains a force acting at a single 
point, provided the work done by this force on the other system of displacements be 
taken mto account. 
The theorem thus becomes 
| | | (Xu! 4+ Yu' + Zw')\aV + | i (Fu' + Go’ + Hw')dS — i | (Fu + G'v + H'w)dS 
= ule, y', 2)X' tule, y', ZY tule, ye, BY 
In order to apply the theorem to the plate problems under discussion, take for the — 
solid a portion of the plate bounded externally by any orthogonal cylinder. Let us 
