﻿S2S Dr. I. J. Schwatt on the 



Aoain 



o 



dry _ d\i dy d 2 u du d 2 y /duV' d*y 

 dx 3 ~ dx z du da? ' dx du 2 "*~ \ax) du 6 



Since d 3 u 2 _ d 2 u du d B u 



dx' 6 dx 2 dx dx 6 ' 



and d z u z _ 9 dhc , a du d 2 u n /du\* 



aW~ =3u dx* +18u dx~dx~ 2+6[ ~ 



IV U/ LIU/ lA/Ui Hub 



therefore 



and 



and hence 



(du\* 



cfu dul rd*u 2 (2\ dhil 

 dx 2 ' \ix~ 2\\_dx' 6 ~V-) U da?\> 



(du\* 1 rd*u* /3\ d*u 2 /3\ 9 dhn 



dx z -da*du + 2\ 



d z u 2 (2\ dhnd^y 

 dx 3 \l) U d?\d^ 2 



d z y d d u dy 



/3\ <*V /3\ 2 dhnd*y 

 -\l) U dx^ + W U dFz]du^> 



+ 3!^ 3 



or written symbolically 



^ 3 



3 I K-l / \ J3 K-a ^K 7/ 



=,? 1 k?o ( - 1) M ^^- • w 



Let us assume that this form holds for all values of n from 

 1 to n inclusive, that is 





We shall then prove that this form holds also for , " a+1 . 

 Differentiating (3) gives 



d n+1 y » 1 ^i -.a/Vv d n u«-« d K+1 y du 



d^ 1 ^=i« ! tt = v W ^ du K+1 ofo? 

 + 

 Now 



» i ^i ^m d r ^ v ~ a i 



1 L£^ l) (-)si.«-3e-j"j 



£ c * 1 / -,x a ,dud n u K -°- ^% 1 / T*ftc\ ^ n+ V" a 



