274 



M. G. Van der Mensbrugghe on the 



It is clear, in the first place, that the force which acts on each 

 point of the thread is a normal to the curve, and comprised 

 within the tangent plane led from this point to the lamellar sur- 

 face ; I assume, further, that this tension has everywhere the 

 same intensity. That being granted, let S be the constant force 

 of contraction of the liquid film, X, p, v the angles it makes 

 with three rectangular axes at any point of the thread which 

 has the coordinates x, y, z; let t be the tension of the thread 

 at this point, and s the arc of the curve sought. We shall ob- 

 tain the following three equations, which express the conditions 

 necessary and sufficient for the equilibrium of the thread : 



d 



S cos X -f 



S COS fJL + 



S cos v + 



('!) 



ds 



<4) 



ds 



ds 



=0, 



=0, 



=0. 



(1) 



dz 



doc du 

 If we multiply these three equations respectively by -x-> -j-, 



> reduce and add the three first members, we shall obtain in 

 us 



this manner, 



« T -x dx , dy dzl < 



♦ [©'.+ ©■♦ (r* 



ds) J ds 



>=o. 



+ 



rdx d*x dy d*y dz dVi 

 ids ' W + ds ' ds* + Js ' I?]*. 



Now the trinomial 



dx 



COS X -— + COS fJL -j- + COS V 



dy 

 ds 



dz 



is evidently zero, for the tension of the liquid film is normal to 

 the curve ; we further know that 



/dx\ 

 \ds) 



+ (!)*+©*-. 



and that consequently 



dx d 2 x dy d 2 y dz d 2 z _ 

 ds ds 2 ds ds 2 ds ds 2 ~~ 



