32 



We shall prove by induction that 



bl 



dai 



m + t + 1 dx 



dbi 



m + t + 1 dx 



> m>t 



[82a] 



[82b] 



We have already shown this to hold for < = 0. Assume it to hold for all ^ < s and for m < n 

 when t = s. We first prove 



n _ 



dx in 



1 (d<~^ dK~^\ 



— ,'?(s-i)(— ^ --j-j 



+ s) \ dy d z I 



By Equation [78a] 



dal -n^s-i) i d^ulzl dH:z\\ ,(.-i) p_ar d_^\ 

 dx {2s -l)2s\dx dy dxdzl 2s \ dy ds I 



Assume Equation [83a] to hold for 



^-^ = L_^(s-1) _^5 2L_ 



dx (m + s) \ dy dz ' 



Then 



dal 



1 d^a' 



dx {m + s + I) Q^2 {m 



r,(s - 1) /d^a 



+ s){m + s + l)\ dx 



2„s-i -,2as-1 



d^t 



dy d X d z 



[83a] 



ri(s-l) ldal-\ dh'-^ 



Ida'-}. db'-l.\ 

 \ dy dz I 



{m + s + 1) \ dy dz 



Therefore, by induction, Equation [83a] holds. Similarly, 



dx in+ s) \ dz dv I 



[83b] 



