52 



APPENDIX 



END POINTS OF A DISTRIBUTION 



An approximate method for determining the end points of a distribu- 

 tion and its trends at the ends will now be described. Let y 2 = f(x) be the 

 equation of the given profile extending from x = to x = 1 ; let m(x) be the 

 corresponding doublet distribution, extending from x = a to x = b. It will be 

 assumed that < a « b < 1 and that a is near 0, b is near 1 . Then m(x) is 

 given by the integral equation 



m(t)dt 



J « [(x-t) 2 + f(x)] 3/2 2 



[in 



Various conditions on m(x) may now be obtained by differentiating 

 [111 ] repeatedly with respect to x. We get 



m(t: 



2r 



T m(t)[2x - 2t + f'(x)] dt = 



■J a j» 



\ (2x - 2t + f') 2 + — (2 + f")l 



T, 5 J 



dt = 



;ii2] 



:i13] 



m (t)15 (2x-2t+f) 3 - 21 (2+f")(2x-2t+f) + f '" (x) 



4r 2r 7 r 5 



When x = 0, r = t and, writing f(x) as a Taylor expansion 

 (x)=ax+ax 2 +ax 3 + ... 



dt = 



:n4] 



[115] 



then also f'(0) = a , f"(0) = 2a £ , f"' (0) = 6a . Now, setting x = in Equa- 

 tions [ill] and [113], we obtain 



dt = -j 



f m(t) 



Ja t 3 



( b B[±l( a - 2t)dt = 

 4 t 5 



[n6] 



:n7] 



[ -^^ - 20a 1 t + 4(4 - a 2 )t 2 jdt = [ll8] 



