THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. 217 
Then to (6) we have in general to add 
‘ole 30 ce ) 
or, otherwise arranged, 
Pas (S.)]+ 40] S|} ne Q} — o[Z,] + El ho |= | \ 
= Pjo[Ny] + Pols) 
The various terms of this expression may be interpreted with the help of the concep- 
tions of sources and doublets. Thus, to go back to (3), we see that the part of U’ 
arising from an element N,ds of normal traction at H has (P)+P,)N.ds for principal 
term. P,+P, is therefore (principal term of the) value of U’ due to a unit element of 
normal force at H. (Since this unit element can only exist in any actual deformation as 
part of an equilibrating system, the phrase due to in the last sentence must be taken 
under reservation. The solution of which P,+P, is the x-displacement at (x’, y’, 0) 
is in fact maintained by a unit element of normal traction at E, acting along with a 
continuous system of force in equilibrium with this element, and distributed over the 
edge in a manner depending only on the statical value of the element, and not at all on 
the position of HK. For any equilibrating combination of elements, the aggregate of 
these continuous systems will disappear.) 
Now the first of the above integrated terms is equivalent to (P, +P,)}— [So]. 
ence 
Hence the discontinuity in 8, at E has the same effect at a distance from the edge 
as would have an element of normal traction distributed over the perpendicular at E so: 
as to give a resultant 2— [Si] 
Again an element — A of normal traction at E, combined with an element A at E’,. 
where EH’=dls, will give 
i d 
U'=A © (P,+Py)ds 
= * (P) +P), if we take Ads=1. 
£ (P+ Py) is therefore due to a unit doublet of normal force at K, and from the term 
4o[N,] we conclude that the discontinuity in N, at E has the same interior effect as. 
a doublet of normal force at H of strength —4o[N,]. 
The other terms may be interpreted similarly. It does not seem possible to account. 
on physical grounds for any except the principal terms of the solution given above. 
The principal terms are of course the same as those deduced in the ordinary theory from 
the * Principle of the elastic equivalence of statically equipollent systems of load.’ 
TRANS. ROY. SOC. EDIN., VOL. XLI. PART I. (NO. 8) 33 
