53 



f 6 m(t]r 35a 3 _ 21Qa 2 t + 60ai (6-a 2 )t 2 + 40(3a 2 -4)t 3 + 24a 3 t 4 ]dt = [119] 



Also assume that m(x) may be expressed as a power series 



m(x) = c Q + c x x + c x 2 + ... [120] 



Then Equation [11 6] gives 



'°l 1 1 \ , (1 1 \ , i b , 1 



or, neglecting l/b 2 in comparison with l/a 2 and setting b = 1 in comparison 

 with l/a, 



c Q + 2c x a(l-a) + 2c £ a 2 log j + ... = a 2 [121 ] 



Similarly, Equations [117]. [118], and [119] give, approximately 



c Q (3a -8a) + 4c a(a 1 -3a) + 6c 2 a 2 (a 1 -4a+4a 2 ) = [122] 



2c Q [5a^-24a 1 a+6(U-a 2 )a 2 ] + 4c 1 a[3a*-15a 1 a+4(4-a 2 )a< 

 +c 2 a 2 [l5a^-80a 1 a+24(4-a 2 )a 2 ] = 



:i23 



3c [35a*-2UOa^a+80a 1 ( 6-a 2 )a 2 +64(3a 2 -4)a 3 +48a 3 a 4 ] 

 +24c i [5aia-35a^a 2 +12a l (6-a 2 )a 3 +10(3a 2 -4)a 4 +8a 3 a 5 ] [124] 



+4c 2 [35a% 2 -252a^a 3 +90a 1 (6-a 2 )a 4 +80(3a 2 -4)a 5 +72a 3 a 6 ] = 



Equations [121 ] through [124] are sufficient in number to determine the un- 

 knowns a, c„ , c, , c . Since the latter three equations are linear and homo- 



1 2 



geneous in c„ , c. , and c , a can be determined from the condition that the de- 



° 1 2 



terminant of their coefficients must vanish. In this way the following equa- 



ai 

 tion of the ~Jth degree in a = — was obtained: 



