Lieber and Yajnik 



f (x.-y, -Ay,t) - f (x,-y, t) 



Lt 



(-Ay)^o (-Ay) 



= - — (x,-y,t) ; 



3y : . , ■ . 



Bf f(x + Ax,y,t) - f(x,y,t) 

 (x,y,t) - Lt 



3x Ax^o Ax 



f (x + Ax,-y, t) - f (x,-y, t) 

 = Lt 



Ax^O Ax 



= — (x,-y,t) ; 



dx 



3f di 



— (x,y,t) = — (x,-y,t) 



We also obtain, by similar reasoning, 



di 



^— (x,y,t) = — (x,-y,t) ; 



oy oy 



3g Bg 



— (x,y,t) = - — (x,-y,t) ; 

 ox oy 



Bg Bg 



— (x,y,t) = - — (x,-y,t) . 

 Bt Bt 



Thus Bf/Bx, Bf/Bt and Bg/By are symmetric, whereas Bg/Bx, Bg/Bt, and Bf/By 

 are antisymmetric. 



Notice that fg is antisymmetric. Also, if f + g vanishes at (x,y,t) and at 

 (x,-y,t), then 



f(x,y,t) + g(x,y,t) + f(x,-y,t) + g(x,y,t) = . 



Hence f and g must vanish individually. 



We shall make use of these properties of symmetric and antisymmetric 

 functions. 



692 



