EXAMPLES OF INDEPENDENT DIFFERENTIATIONS. 109 



3. If u = Aaf-y -' + Ex^y? + Cx^f + ... 

 where a + a' = + /S' = 7 + 7' = ... = , 



du du 

 shew that 



# -=- + y -=- = nu. 

 ax J ay 



In this example u is called a homogeneous function of n 

 dimensions. 



4. If u be a homogeneous function of n dimensions, 

 shew that 



d*u d\i . . du d'*u d\i . du 



5. If u be a homogeneous function of n dimensions, 

 shew that 



d?u d z u d"u . . 



6. Verify the theorems in Examples 3 and 4 in the follow- 

 ing cases : 



u = (x + y}\ 



XV 



u = , 

 x + y 



7. If u = a?V + e'yV + atyV 2 , shew that 



d 4 u 



dx l dy dz 

 8. If u = e* 9 ', shew that 



Syz. 



9. If w = y V (a 2 - # 2 ) + a; V ( 2 - f) , shew that 



dudu , ., 2 ax // 2 2N / d? u \ 2 * 



T~ j + V ( a x } V ( a SH j j ) = -77-8 . A ,/ 2 ^ 



da; </ ^ ' \dx d) irar v ) 



