54 Mr. A. M. Worthington on Calculating Surface-Tensions 



It is from this equation that Prof. Quincke calculates the 

 value of T. It is evident that in neglecting the curvature of 

 the vertex we are neglecting the pressure due to this curva- 

 ture transmitted to the whole area K — k. Thus, if b be the 

 radius of curvature in question, the pressure disregarded is 



2T 



— (K— -k). The surface-tension T has to balance this as well 



as the hydrostatic pressure due to the w r eight of the liquid, 

 and neglect of this term will lead to too small a result. 



Again, in neglecting the curvature in the plane at right 

 angles to the plane of the diagram, we evidently leave out of 

 account the tension exerted along each edge AE, DE', of the 



T 



slab, which produces a pressure ^ on each unit area of the 



surface. 



Since the surface is one of revolution, B/ is the length of 

 the normal intercepted by the axis, and writing cj> for the 

 inclination of the normal to the axis measured on the side of 

 the vertex, i. e, for the edge-angle of the drop- or bubble- 

 forming fluid at any horizontal section, we have ^ = -, 



where x is the horizontal radius of the section ; and the 

 pressure on a horizontal strip of the rectangular end of ele- 

 mentary depth dz is ^— and the total action omitted, 



PK-k 



I T sin <p dz . 



Jo » 



x 



so that the complete equation is 



T = v j 1 — +y(K-*)-T I sin0<fe 



Jo x 



(2) 



The value of the integral of the last term to a first approx- 

 imation is shown by Laplace (Mec. Celeste, livre x. 2 e Suppl. 

 p. 483), or by Mathieu (TMorie de la Capillarite, p. 137) to be 



. 1 — cos 3 % 



4a 2 



3 x ? 



/T 

 where a = / W ^* 



When x is equal to the maximum radius L, then = 90° and 



