1888.] Mr G. H. Bryan, On the Stability of Elastic Systems. 205 
idea of the greatest thrust which can be applied along the ends of 
a very broad thin flat piece of clock spring, without causing it to 
double up. 
To prove the above result take the axis of y along one edge of 
the strip, the axis of # being in its plane perpendicular to the 
edge, and that of z perpendicular to the surface. Let the plate be 
deformed so that any point of it receives a small displacement 
w, independent of y, perpendicular to the plane of the plate. Then 
if s is the length measured along the new middle surface of the 
plate in the plane of xz, the work done in stretching the surface by 
the thrust P per unit length of the strip is* 
t /ds sus ‘ dw|? 
while the potential energy of bending per unit length is 
tw? 
Crp | ale wae 
fee Gar da? 
Hence the plate will be stable if 
L day” 
wf S day 1) ae< guit (—™ \f 
m+n/Jo dx 
for every possible deformation. 
If the edges are fixed in position w=0 both when #=0 and 
when #=J, and therefore w may be expanded by Fourier’s series in 
the form 
If on the other hand the directions of the tangent planes along the 
edges are fixed, = =0 at either end, and w may be expanded in 
the form 
= 1x 
w= >", a, COs ee (14). 
In either case the eondition for stability (12) requires that 
r=0 ™m y= 00 rar 
LPSI~? at < 4nh! (ee) sre ae ae, 
2 Safa? 
or P< §nh? (—" \ ee peta 3 tsi (15) 
UI) ao 
for all values of the constants a,. 
* Compare Lord Rayleigh, Sound, Vol. 1. p. 136. 
Pp y 7 
15—2 
