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considerations, and found the problem to be much simpler than 
had been generally supposed, and that the three necessary and 
sufficient equations for the equilibrium of an element of the 
surface could be obtained in a very simple manner. 
By the principles of statical science, when three forces are in 
equilibrium at a point they must be in the same plane, because 
the resultant of any two must be equal in magnitude and oppo- 
site in direction to the third force. We know therefore that 
when two opposing tensions and an-external force are in equi- 
librium at an elementary area of a flexible surface, the same 
rule must hold good; and it also must hold good if the tensions 
transmitted in any manner through a sheet are equivalent to 
resultant tensions acting in different directions through the ele- 
ment of the surface, for each set of resultant tensions and their 
corresponding portions of the external forces. From these con- 
siderations we learn, that in all the ordinary problems of the 
equilibrium of flexible surfaces, of regular forms and symmetrical 
positions, where the. external forces arise from gravity or the 
pressure of fluids, the tensions will act along the lines of curva- 
ture of the sheet, since it is only for poimts taken in succession 
along such lines, at right angles to each other at each point, 
that consecutive normals to the surface meet, and the conditions 
can be satisfied, 
If s and s’ are ares of the lines along which the resultant 
tensions act, measured from any fixed points; ds and ds’ 
elements of these arcs at right angles to each other, forming 
the sides of an elementary area ds.ds’ upon the sheet, of which 
the thickness at this element is r, and density p; then prds. ds’ 
is the mass of theelement. Let X, Y, Z be the external acceler- 
ating forces, acting parallel to the axes of coordinates respectively 
upon the elementary mass, Let 7’ be the tension due to a unit 
of breadth, acting in the direction of s upon the element, 7” that 
acting in the direction of s’; with dz, dy, dz the components of 
ds in the axes respectively ; and da’, dy’, dz’ those of ds’. 
