462 RICE ART. K 



and therefore 



Xy ban dx'dy'dz' = — (Xy 8x) dx'dy'dz' 



dy' 



- ff 



'dX 



Y' 



T 8x dx'dy'dz' 



J J dy 

 = U'iXy 8x) Ds' - j I j-^ 8x dx'dy'dz', 



and so on. When we make the substitutions in the first integral 

 of [367] justified by these transformations, we convert equation 

 [367] into the form [369]. It might be as well to write the 

 first integral in [369] in full for the sake of clarity; it is 



f f f ( /dXx' dXy dXz'\ 



/dYx' dYy dYz'\ 

 -^\^ ^~By^^^F)^y 



, /dZx' dZy dZz'\ \ , , , 



where of course 5a:, dy, Sz are to be regarded as functions of 

 x', y', z', infinitesimal in value. Similarly the third integral 

 written in full is 



/{ (a'Xx' + ^'Xy. + y'X,,)8x 

 -\-(a'Yx' + /3'Fk' + YYz')5y 

 + (a'Zx' + ^'Zy + yZzO^z }Ds'. 



We shall neglect for the moment the point raised at the bottom 

 of page 189 concerning surfaces of discontinuity, returning to it 

 when we give a proof of Green's theorem, and proceed with the 

 general fine of development. Taking the result [369] we shall 

 rearrange it so as to collect all the terms involving 8x, all those 

 involving dy, all those involving 8z and all those involving 8N'. 

 It is then written in the form 



