Transition Layer of a Liquid on its Surface lension. 881 



and similarly 



A 3 = (« 3 +a± +85 + • • • a n ), 



and so on. We then have 2X nx =A n , and from this equation, 

 the special case 2\ x =A 1? and equation (8), we obtain 



««=£; (10) 



It is necessary next to calculate the values of 



a \) a 2) a 3i • • • a n> 



The component of attraction at right angles to the plane ab of 

 a molecule lying in the plane on a molecule in the slab B at 

 the distance nx from the zero of coordinates is given by 



-g — ■ , where z denotes the distance of separation of the 



molecules. The coordinates of the latter molecule are 

 (nx, 0, 0), and if those of the former be denoted by (rue, wx, 

 vx) we have 



*= ^(nW + ivV + vW), 



and the above expression becomes 



hi 

 x 5 {n 2 + w 2 + v 2 } B ' 



The component of attraction/^ exerted by all the molecules 

 in the plane ab is therefore equal to 



h /% xl ^ x hn , .•a" hn 



+ 4 i * Sf 2 , O , 9-13 +' 



(nxf t * w Zi 7=1 x 5 {n 2 + ic* + v 2 Y Zi^i^ + v"? 



The work a n required to move the molecule to infinity is 

 given by 



+ 2 



On developing the right-hand side of this equation into a 

 series (we need not retain the terms involving values of 

 v and w higher than 3), the value of a n can readily be cal- 

 culated for different values of n. The equation gives 



k 

 x\n 2 + v 2 j*' 



a x 



= ^•7532, a 2 =- 4 -15195, a 3 = T4 "05056, and a 4 = -j -0219. 



It is not necessary to obtain values of a n of a higher order 

 than the fourth. 



