Permanent Forms of Mathematical Expressions. 125 



Qi 



d 



~ d v 



d d d 

 dz dv dw 



d d 



z du y y du z 



d d 



d d 

 z dw y y dw z 



d d d 



+ W *dv- + W ^ + W ^ 



x dw„ y dw„ z dw z 



d d n d d d 



+ U -d^ y - U ^ +2u ^ y + ^-^d^ z - 2U ^ z 



d d d d d 



+v -av xy - v ** m + 2v »>a\y + ^~ v ^dV yz -*°»dr m 

 + w ^ xy ~ w *»ir xz + 2w ^ + ^ - w ^ z ~ 2Wyz ck* 

 + w -£ + "«4; + w -£ +w »4; + w -4 +w -*7" 



d d d d d d 



~ Vxx d^ x ~ v *"dw^, ~ v "dw m ~ Vyy d^ y ~ Vyz dw^ z ~ Vm dm m 



d d 



It is easy to find examples of the results which flow from 

 these conditions ; for example 



(a) There is no linear scalar function of the second 

 differential coefficients of (p which satisfies the conditions of 

 permanency of form except 



dfy dfy dty 

 doc 2 dy 2 dz 2 



