Mr. J. T. Riley on Capillary Phenomena. 197 



TT 



= -5 , where z is the height of the point considered (rnf, fig. 5) 



from the axis of x taken in the level surface. From the 

 differential equation of the curvature at the point w! we get 



dx 2 l^ __ g pz 



Ml)?"" 



and, integrating, 



H goz 2 



= **k- + const. 



^*m 



*\* 2 



dz 



When = 0, -j- = 0, and const. = — H, so that the equation 



becomes 



dx 



HI 



1 %/wir 2 



=+>/ 



Introducing the angle j3 made by the tangent to the surface 

 with the ordinate z, we get finally 



i-(l-sin/3). . . . (1) 



99 



Now let the surface at P make an angle a with the surface of 

 the plate, at Q an angle a! . The horizontal components of 

 the tensions will be H sin a and H sin a! respectively. The 

 hydrostatic tension on the plate from S to Q will be 



i 



gpzdz= * rx ~ =H(1— sina') 



from formula (1). 



Similarly the hydrostatic pressure from S to P is 

 H(l — sin a); supplying these values in the equation for equi- 

 librium we have the identity 



H = H. 

 We thus see that the resultant forces acting on the plate are 

 two equal and opposite tensions which will tend to pull the 

 plates asunder. In every case the identity shows that the 

 magnitude of these tensions is just the same as if the surfaces 

 came up perpendicular to the plates without elevation or 

 depression. 



In conclusion I would point out that the mathematical 

 theories of Laplace, Gauss, and Poisson do no more than 



