650 



ON THE COHESION OF VLVIDS. 



the same is true of a surface of simple 

 curvature; but where the curvature is 

 double, each curvature has its approjjri- 

 ate effect, and the joint force must be as 

 the sum of the curvatures in any two perpen- 

 dicular directions. For this sum is equal, 

 whatever pair of perpendicular directions may 

 be employed, as is easily shown by calculat- 

 ing the versed sines of two equal arcs taken 

 at right angles in the surface. Now when 

 the surface of a fluid is convex externally, its 

 tension is produced by the pressure of the 

 particles of the fluid within it, arising from 

 their own weight, or from that of the sur- 

 rounding fluid ; but when the surface is con- 

 cave, the teiK^ion is employed in counteract- 

 ing the pressure of the atmosphere, or where 

 the atmosphere is excluded, the equivalent 

 pressure arising from the weight of the par- 

 ticles suspended from it by means of their 

 cohesion, in the same manner as, when water 

 is supported by the atmospheric pressure in 

 an inverted vessel, the outside of the vessel 

 sustains a hydrostatic pressure proportionate 

 to the height ; and this pressure must re- 

 main unaltered, when the water, having 

 been sufficiently boiled, is made to retain its 

 situation for a certain time b}' its cohesion 

 only, in an exhausted receiver. \'\'hen, 

 therefore, the surface of the fluid is termi- 

 nated by two right lines, and has only a 

 simple curvature, the curvature must be 

 every where as the ordinate ; and where it 

 has a double curvature, the sum of the cur- 

 vatures in the different directions must be as 

 the ordinate. In the first case, the curve 

 may be constructed by approximation, if we 

 set out from a point at which it is either ho- 

 rizontal or vertical, and divide the height 

 into a number of small portions, and taking 



the radius of each proportional to the recipro- 

 cal of the height of its middle point, above or 

 below the general surface of the fluid, go on 

 to add portions of circles joining each other, 

 until they have completed as much of the 

 curve as is required. In the second case, it is 

 only necessary to consider the curve derived 

 from a circular basis, which is a solid of re- 

 volution; and the centre of that circle of 

 curvature, whicii is perpendicular to the sec- 

 tion formed by a plane passing through the 

 axis, is in the axis itself, consequently in the 

 point where the normal of the curve inter- 

 sects the axis : we must therefore here make 

 the sum of this curvature, and that of the 

 generating curve, always proportional to the 

 ordinate. This may be done mechanically, 

 by beginning at the vertex, where the two 

 curvatuies are equal, then for each succeed- 

 ing portion, finding the radius of curvature, 

 by deducting the proper reciprocal of the 

 normal, at the beginning of the portion, 

 from the ordinate, and taking the reciprocal 

 of the remainder. In this case the analysis 

 leads to fluxional equations of the second or- 

 der, which appear to afford no solution bv 

 means hitherto discovered ; but the cases of 

 simple curvature may be more easily sub- 

 jected to calculation : the curvature varying 

 always as the ordinate, the curve belongs to 

 the general description of an elastic curve. 



111. ANALYSIS OF THE SIMPLEST FORMS. 



Let the greatest ordinate of the curve (AB, 

 Plate 15. Fig. 113.) be called a, the arc of the 

 circle of curvature at the vertex (AC) z, and 

 let us suppose.that whilethiscirde is uniformly 

 increased, the curve (AD) tlows with an equal 

 angular velocity, then the fluxion of the 

 curve, being directly as the radius of curva- 

 ture, will be inversely as the ordinate y, and 



