Tension of Vapours near Curved Surfaces of their Liquids. 383 



Let, then, the equation of the surface referred to its tangent 

 plane be 



where a and b are consequently the principal radii of curvature 

 at the point, and the higher powers of x and y are neglected ; 

 for it is obvious that it is only points near the origin that are 

 of any importance. Now z is to be very small, so that x 2 and 

 y 2 are small compared with a and b. Let r be the radius 

 drawn from a point situated on z at a very small distance 7 

 from the origin to any point on the surface, and let 6 be the 

 angle between this radius and the normal at the point where 

 it meets the surface, and yjr the angle between this normal 

 and z. A molecule then emitted from 7 in the direction r 

 will have to travel a length r within the liquid before escaping. 

 Let then f(r) represent the numbers emitted after having tra- 

 velled this distance within the liquid. All that we know of 

 f(r) is that it vanishes for all values of r above a very small 

 quantity. Hence we may express the numbers emitted from 

 this point by the double integral 



') cos 6 dx dy 

 * 2 COS y{r 5 



and bearing in mind that a and b are large compared with such 

 values of r, x, y, and 7 as do not make /(f) to vanish, we can 

 evidently expand this into the form 



where A , A 1? B x are functions of x, y, and 7. It is now to 

 be observed that, on account of the symmetry of the equations 

 involved in x and y, we must have 



\\ A x dx dy —\\ B x dx dy, 



and that, consequently, the numbers escaping from this point 

 may be expressed as 



=JP 



-Xf(- 



(hi)- 



n = n + n ± i - + 



and if this be integrated for all depths y, as a and b are the 

 same for each the result must be of the same form, and may 

 be written 



N=N » +N 4 + &- 



In this N is the number that would be emitted if the surface 

 were flat; and by changing the signs of a and b we get the 



