350 PROCEEDINGS OF THE AMERICAN ACADEMY. 



D w (w + *>) = D w u -f JD w v, D w ii n = n • u n ~ l ■ D w u, 



D w (« • *>) = v • D w u + m • D w v, U w [-j= ^ > 



2>»/(«*) =/'(«)• A>(«)- 



The normal derivative of ^^ with respect to v is a scalar function 

 which, if differentiable, has a normal derivative with respect to v, and 

 since by definition 



V v * = ^4, (27) 



I'V 



n , i ( aA„ a« a^ a« a*„ a?' ) , . 



/* v - ( d^r dx dy dy dz dz ) 

 we may write 



i \d 2 u /a^v awaA 2 awaA 5 

 2 j a 2 ^ a« a*y a 2 « ay ay a 2 « a^ ay 



V \dx-dy dx dy dy • as dy dz dz • dx dz dx 



+ ^7 1 Or W 3^ ' dv J + dy\ dy dy' dv) 



,d_u(dh_dv MA ) ( 2 a\ 



^ dz \dz dz' dv J j V ' 



1 \ d 2 u dw dv d 2 u dw dv d 2 u dw dv ) 



DwDvU = I7^ 2 \d^'d^'d^ + dy'^'d^ + d^^'dz\ 



d"u (dw dv dw dv\ d 2 u ( dw dv dw dv\ 

 dx ■ dy\dy 'dx + dx'dyj + dy dz\Jdz ' dy dy'dz) 

 I dw dv dw dv\ 

 \z\dz dx dx dz) 



llv* • tlw" 



d 2 U 



dx ■ a. 



1 j du(d 2 v dw d 2 v dw dv dw\ 



+ h 2 ■ h w 2 \ dAjti 2 ' dx~ dydx ' dy + dx ■ dz ' dz J 



2 du dv(dh v dw dh v dw dk v dw\ ) 

 ~ h v dx dx\dx dx dy dy dz dz J ) 

 1 ( dufdH dw d 2 v dw dv dw\ 



hFh? \ d~y W'¥ dz-dy' dz + dydx'dy) 



2 du dvfdkv dw dhv dw dk v dw\ ) 

 ~h v '~dy" dy\dy "dy Hz" dz dx' dx J ] 



