468 Lord Rayleigh on the 



as conceived by Gauss. This principle, applied to a hypo- 

 thetical displacement in which the point of meeting travels 

 along the wall, leads with rigour to the required result. 



In view of the difficulties which have been felt upon this 

 subject, it seems desirable to show that the calculation of the 

 angle of contact can be made without recourse to the principle 

 of interfacial tension or energy. This indeed was effected by 

 Laplace himself, but his process is very circuitous. Let 

 OPMbe the surface of fluid (crj resting against a solid wall 

 N of density a 2 . Suppose also that cr 3 = 0, and that there 

 is no external pressure on M. At a point M at a sufficient 

 distance from the curvature must be uniform (or the 

 potential could not be constant), and we will suppose it to be 

 zero. It would be a mistake, -p. -.^ 



however, to think that the sur- 

 face can be straight throughout 

 up to 0. This we may recog- 

 nize by consideration of the po- 

 tential at a point P just near 

 enough to for the sphere of 

 influence to cut the solid. As soon as this occurs, the po- 

 tential would begin to vary by substitution of <r 2 f° r °"i> an( ^ 

 equilibrium would fail. The argument does not apply if 



We may attain the object in view by considering the 

 equilibrium of the fluid M N 0, or rather of the forces which 

 tend to move it parallel to N. Of pressures we have only 

 to consider that which acts across M N, for on M there is no 

 pressure, and that on N has no component in the direction 

 considered. Moreover, the solid o- 2 below ON exercises no 

 attraction parallel to ON. Equilibrium therefore demands 

 that the pressure operative across M N shall balance the hori- 

 zontal attraction exercised upon M N by the fluid o^ which 

 lies to the right of M N. The evaluation of the attraction in 

 such cases has been already treated. It is represented by 

 MN . 0"i 2 K o , subject to corrections for the ends at M and N. 

 The correction for M is by (41) <r 1 2 T (2 sec 0-cos 0), and for. 

 N it is o- 1 2 T . On the w T hole the attraction in question is there- 

 fore 



o-^M N . K -2T sec <9 + T cos 0-T o }. 



We have next to consider the pressure. In the interior of 

 M N, we have o" 1 2 K ; but the whole pressure M N . ct^Kq is 

 subject to corrections for the ends. The correction for M we 

 have seen to be 2cr 1 2 T sec 6. In the neighbourhood of N the 

 potential, and therefore the pressure, is influenced by the 





