147 
Then the component tensions acting on the element in the 
directions of the axes respectively, will be 
, dx , dy 4 ’ dz 
Lids. = Tas’. = Tds' . 7 
for Tds’ the tension acting on the element in the direction of s ; 
and 
Ly nace 
ds. aa) Ids. 73 
those for T’ds. 
. Now the variations of these in passing from one side of the 
element to the opposite side will be the only internal forces 
entering the equations of equilibrium; and the three necessary 
and sufficient equations for the equilibrium of the element under 
the action of the external and internal forces become as follows, 
a(ras.) a t’as. | 
ATp.ds.ds’ + Aa 3 ts 1) boa Gb), 
a( Tas’. a a(T'as. | 
Yrp.ds.ds' + —~——=. ds + aR OS = Osennce 2) 
a( Tas’. rl a(T'as.%) 
PEE ds es 2 OGM 
ds iAnadtisan Paib 
which are applicable generally to the cases of the equilibrium of 
flexible surfaces. 
When the external fsreey is anormal pressure V ona unit of 
area, such as the pressure of a fluid, and R, R’ are the principal 
radu of curvature at the element, then by resolving in the 
direction of the normal and performing the differentiations, we 
obtain the well known formula NV =74 = whether 7’ and 7” be 
constant or variable since the coefficients of = and at 
ds’ disap- 
pear. 
