146 Mr Love, Note on Kirchhoff’s theory of the [Nov. 28, 
If 1, m, n be the direction cosines of the outward-drawn normal 
to F(x, y, z)=0 the surface tractions are 
F=(P4+mU+4+nT 
G=l1U+mQ+n8 
H=1T+mS + nk. 
Let us now take x= ew’, y=ey’, z=ez' and substitute in the 
equation F(a, y, z) =0, we obtain an equation which may be written 
F' (#, y', 2) =0 where F" contains only finite constants. Let 
ein je 2 ue and J?4+ mm? +n? =1. 
OE ORT | OZ 
i ,__ Ou , Ow Ov 
Write also T agi oo meg agin? 
and let P’... be the functions formed with the e’... in the same 
way as P... are formed with the e..., the equations of equilibrium 
become three such as 
CREM CURIOUS ine 
ae ay ae +epX=0, 
where p is the density; and at F’=0 we have three surface 
conditions such as 
UP +m U4+nT =eF. 
The solution for wu, v, w consists of two parts, viz. that which 
corresponds to the bodily force when there is no surface-traction 
and that which corresponds to the surface-traction when there 
is no bodily force. Of these terms the former are of the order 
epX, and the latter of the order ef. Thus the displacements 
depending on the bodily forces are negligeable compared with 
those produced by the surface-tractions, and we may obtain the 
internal condition of the indefinitely small body by omitting the 
bodily forces. 
Similar considerations show that if (2, y, 2) always refer to 
points within an element of a body all whose dimensions are of 
the order ¢, the strains ¢... are great compared with the dis- 
placements w, v, w. 
By a thin plate is meant a mass of elastic material which 
in the natural state is bounded by two parallel planes and a 
cylindrical surface cutting them at right angles, the distance 
between the planes being very small compared with the least 
linear dimension of the space enclosed within the bounding curve 
on either of them. 
