Desai and Lieber 



of the preceding equations can be represented in the form of a power series. 

 However, the power series solutions need not be expansions in powers of r. In 

 fact, it is advantageous to use a transformation which affects a contraction in 

 scale. We then have two advantages in the computation of these analytical ex- 

 pressions: the first is increased precision, and the second is that there is a 

 less number of terms to compute for a given value of r, because of the in- 

 creased convergence. The application of the boundary condition at the wall is 

 simplified if the transformed coordinate varies from to co when r varies from 

 1 to CO. The following class of transformations which affect a logarithmic con- 

 traction in scale has this property. 



r = e^^''^"^^ 1 < r < CO 



r = e^^ '■^ < s < CO 



(2.73) 



and so on, c being a constant scale factor. At first, the transformation r = 

 e*^^ was used, but when it was found that higher precision was needed in the 

 calculation of the second iteration, the transformation r = e^*'''^"^) was used. 



The results presented in this paper are obtained by using the later trans- 

 formation r = e*^*^"^"^). The equations obtained by using these transforma- 

 tions are contained along with relevant algebraic details in Ref. 2. These equa- 

 tions are then solved by expressing the solutions as sums of power series in 

 the variable s. We thus obtain 



'i(^) = E "^i 



j=i 



e ■'CI „cs y-i 

 k= 1 



KlJw^ 



k= 1 



K13 J 



+Y1, e^ + Y1 e~^ 



5 6 



(2.74) 



B,(s) -- ^ YI3 

 j = i 



e "c ,„cs 



» MIO, J 



E 



;k _ „-2e 



E 



L ok 



+ ¥1,6^^ + Yl, 



(2.75) 



8 / CO \ " 



Di(^)= E Y2JE DV-'"M+ E Dlk"^^*^"^ 



j=l \k=l J k= 1 



(2.76) 



j-l \ k= 1 



D^cs) = J] Y2. f; m^'s^-A, J2 ^v^ 



,k-l 



(2.77) 



572 



