186 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
Similarly in I. (v) the multipliers of P,,,, Z, are displacements compounded of 
the types (iii), (v), with sign of m changed, and so on. 
The full explanation of this peculiarity will be given presently, when it will f 
seen that an independent verification of all the results may be obtained by means of 
the important principle known as Betti’s Theorem. 
In the meantime we may examine the scheme of solutions from another very 
important point of view. 
With reference to any individual solution, the following questions are obviously of 
prime importance :— 
(1) What is the resultant stress transmitted ? 
(2) Is the whole potential energy of the part of the solid bounded internally by a 
given cylindrical surface, finite or infinite ? 
Now, in order to single out those solutions which convey a finite resultant stress 
across any cylinder (or other surface) surrounding the origin, we have merely to look 
at the table of coefficients. Thus, for instance, I. (i) appears with coefficient X,/327uh, 
from which we may infer (as verified below) that this solution conveys a stress with 
resultant a force of 327“ units parallel to the axis of w, and passing through the 
origin. 
In this way we find that the six solutions, corresponding to the six elements when 
specify the resultant of a force system, are 
LE COMED), (CaN atl lO aOL (Gu) (oo). (crabs 
For these we shall write down the values of the stresses pp, po, pz, the components — 
of the stress across the cylinder p = constant. 
In all, of course, we have 2z=0, and in I. in addition %=zo=0. 
i 5 ron a-15 
lle (1) Hai 5} “pltta—32p ~*) 008 w 
2 +1 
ae = '+a—32p*)sino 
The resultant is a force along Ox, of magnitude 
i Iie COS w — pw Sin w)pdwdz, taken over the cylinder p, 
_fa-15 atl 5 
= € eas 5) aa 2h - Qu = — 32rph. 
i (it) i (¢ eS +a—8 #p-*) sin w | 
age eee 
Du = - ae eo) 2p) cos w | 
The resultant is a force along Oy of magnitude — 327uh, 
L. (vi), m=0. Rabe ee aad 
w= 0 | po = 2up’. 
