1888.] Mr G. H. Bryan, On the Stabihty of Elastic Systems. 201 
Employing as usual P, Q, R, S, T, U to denote the stress com- 
ponents in the position of equilibrium, X, Y,Z the bodily forces, 
FG, H the surface tractions 
db = Pde + Qof+ Rdg + Sda + TSb + Use. ase (5), 
and since ¢(e, f,g,a, b,c) is a homogeneous quadratic function of 
6, f, 9, a,b, ¢ 
od = 6Pde + 8Q6f+ SRdg + 5Sda + STS + 6USe 
=D (OE Of ON OF LOU WOO ps em eh s Neate “iceipuisiasia = (6), 
and is therefore, we know, essentially positive. Since V is a func- 
tion of #+u, y+, 2+, therefore 
OV OV OV 
bV= ua + 8u ay + dw wa =— Xbu—Vbu— Zou... (7), 
Ra gi Vi 50 nos 
&V =du An aa + Ov" oa a= +6 a 
o7V OV o’V 
TBO E I ay ve + 26w du vn + 28ud0 = Hey 
=O AGOU 10 YOU OL OU aya oic si neisio ies suiriesiole sis sce os cine (8), 
lastly, SPs = 1900 Choy Ja l0 scsebee soreckooeaee (9), 
&T =— 6Fbu—6Gdu—sH bw ............ (10). 
3. Since 6’¢ is essentially positive the inequality (4) can only 
hold if the sum of the second and third terms is negative and 
numerically greater than the first, and therefore @ fortiori cannot 
hold if they are small compared with the first. From this we may 
shew that the displacement must in general be such that the 
strain variations (6e, of, 6g, 6a, 5b, dc) are infinitely small com- 
pared with the displacement variations (du, dv, dw). 
For suppose these variations to be small quantities of the same 
order. Then in order that i | | po Vdadydz may be comparable 
27, 2 
in magnitude with [[]®eacayae, Ae ser ae ... must be of 
magnitude comparable with the elastic constants m, n, and this 
OV MCOVe OV. 
Ox d oy ? Oz 2 
as appears at once by integration. But from the equation of equi- 
librium (3), forces of this magnitude will produce finite strains in 
the body instead of infinitely small ones. As above stated, this 
is possible only for a few very extensible substances like jelly. 
must necessarily also be the case with the forces — 
