140 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
Differentiate the first of equations (i) with respect to s. Thus 
‘t) dd Bs de Uf, sone U 
ds = (Z eo eae @ +sine 7 pane Gadel Sas ay 
a 2 
= cos ¢( ~sin ey ete. 
d df_ ay idedf 
ds dn dxdy ds dy 
Gi dadj Slay 
- dzdy ds dn ods. : . . . Gn} 
and at O, 
Similarly from the second of (i), 
Of af aa df 
@f_@f 1 df a 
or ay do aan - aot) 
dq? 2 F ; 
Thus the values at O of %% 2% 2 are known when / and ~ are given along 
dx’ dy dady’ dy? 
the boundary. 
af, OF ; pe 
de® ae v/ being an invariant for all systems of rectangular axes, we may also 
conveniently take 
oe Lid ae 
ee 
im 
4 
a 
1. Equations of equilibrium. Form of solution for a plate free from bodily force. 
The equations of equilibrium of a homogeneous isotropic elastic solid are of the 
form 
==! 4 Xx = © 
dx dy " dz x 
— Y — 
da dy = dz a : (1) 
d Le d Ye d 2z 
pale a i Ly = 0 
dx dy dz : 
— 
where X, Y, Z are the components of the bodily force per unit volume, and zx, yy, %, 
xy, «z, yz are the components of stress, these being given in terms of the displacements 
u, v, w by the equations 
du & Ce =) 
dy dé 
8 
8 
I] 
“> 
cb 
+ 
i) 
as 
S 
= 
PS du du 
= \A+2p : a c df - D) 
ck i My tas Mae + de (2) 
a dw = dv du 
zz = AA+2 : = e ) 
i es Me * dy 
where 
Aves au dv dw 
de + dy * & 
