
EQUILIBRIUM OF ELASTIC SOLIDS. 109 
To find an expression for the curvature produced in a flat, circular, elastic 
plate, by the difference of the hydrostatic pressures which act on each side of it,— 
Let ¢ be the thickness of the plate, which must be small compared with its 
diameter. 
Let the form of the middle surface of the plate, after the curvature is pro- 
duced, be expressed by an equation between 7, the distance of any point from the 
axis, or normal to the centre of the plate, and x the distance of the point from the 
plane in which the middle of the plate originally was, and let @s =/(dz) + (dr). 
Let %, be the pressure on one side of the plate, and 4, that on the other. 
Let p and q be the pressures in the plane of the plate at any point, p acting 
in the direction of a tangent to the section of the plate by a plane passing through 
the axis, and g acting in the direction perpendicular to that plane. 
By equating the forces which act on any particle in a direction parallel to 
the axis, we find 
dr dz dp dz 
ad? x 
“Pa, ge Ti tlrp Ts $47 (hy hy) So = 
By making p=0 when r=0 in this equation, 
Tr As 
fae Dt dea hy) . . . (51) 
The forces perpendicular to the axis are 
dr\? dp d 
tp (FZ) +tr ee _ P= (hy —hy) » S29 t= 0 
Substituting for p its value, the equation gives 



_ (hy = hy) (F dr =) (4, —hy) 5 (dr ds d?x ds d*r ; 
(ipRPERL Taide de) "J Def = dz ds? dx a rae 
The equations of elasticity become 
ene =(—- su) (p+o+ Bf p 
Mw pam) Ce = 
or 1 h, +h, q 
oa ae Fn) (v+9 Maga Ses 
Differentiating BD sa ( r) , and in this case 
d dr\r 
ddr _ dr dr dos 
dr ds ds ds 
2 dor 
By a comparison of these values of ap 
dr 1 h,+h gq arp dp d 
1- ese) 123 
( 2) 5m) (p+a+ 2 \+ fr be m +r(gg- sa) (+ dr 
PB dig? da sy 
leo Th pe ee 
