Tension of Liquid Films. 275 



The above equation reduces itself to 



dt n 



whence 



t = constant, 



a formula which is the expression of this general law : — 



1. Whatever be the laminar surface on which is placed the 

 flexible thread, the tension of the latter is everywhere the same. 



This result permits us to write the equations (1) under the fol- 

 lowing simple form : — 



d 2 x a 



t -=-g = — o cos A, 



t-j^~-S cos fi, 



t-r§ = — Scosv. 

 These three relations, squared, and then added together, give 



<*[©)'+©)'+©>- 



Now if p represents the radius of curvature at any point of a 

 curve, we have always, by a theorem of the differential calculus, 



//A\ ! , /dW /d*z\* ' 

 V(tf) + (tf) + W) 



hence we have also 



- = S 2 • 

 p 2 ' 



whence, disregarding the sign, 



P 



From this follow two other laws, which are very simple and 

 completely general : — 



2. The curve assumed by a flexible inextensible thread without 

 weight, subjected to the action of the force of contraction of a liquid 

 film in equilibrium, has everywhere the same radius of curvature. 



3. The ratio between the tension of the thread and the radius of 

 curvature is constant and equal to the force of contraction of the 

 liquid film. 



T2 



