194 ME. GEOEGE W. WALKEE ON THE INITIAL ACCELEEATED 



when r = a, or 



1 3 J k x 30 _ 301 



when r = a. 



In performing the partial differentiations, must first be expressed as a function of 



independent variables r and /A = - , and after differentiating with respect to r, the 



functions must be expressed in terms of independent variables x and CT, and the 

 differentiation with respect to w performed. 

 Let 



0oce a ~ ' et ', and put X = rau/a- 



The surface condition at r = a may now be written 



i a [ p,,/,] 



J- U \ /I / UU/ /-, 



-= \ XGxdi T ^- } = 0. 

 ns cis [ dr J 



The simplest form for is 



e -6p T 2 



= - where p 2 = - r-. ,+y 2 + z 2 = r 2 +XV, 



p 1K 



and derived forms are obtained by successive differentiations with respect to x. 



We shall now approximate by neglecting terms involving higher powers of X than X 2 . 

 Let 



where 



A] is of order X, 



A a is of order X s , 

 and higher terms are neglected. 



Now 



<o = 

 where 



-- - - . 



\r or/ r 

 Thus, as far as squares of X, we have 



where 



fji = x/r. 



On performing the differentiations, as already explained, we find that the surface 

 condition is 



when r = a. 



A , a2 . . N A fi)2 . 



- A (^u-r//!)- A 2 (0-tyi-3/r a ), 



= 0, 



