468 RICE 



ART. K 



at the points A and B respectively. Also if a^', 13/, Ja and an', 

 ^b', Jb' are the direction cosines of the outward normals to s' at 

 A and B, respectively, then u/Dsa' and —cxb'Dsb' are each equal 

 to the sectional area, since the sectional area is equal to the 

 projection of either of these sections by the surface on the plane 

 OY'Z', and a is the cosine of the angle between the normal to an 

 element of the surface and OX', which is normal to OY'Z'. 

 (The figure shows that the minus sign is necessary in one of the 

 results, since in one case the normal directed outwards will 

 make an obtuse angle with OX'.) Hence the result of integrat- 

 ing (d(j)/dx')dx'dy'dz' throughout the part of the region within 

 this column is equal to 



aA(i>ADSA + aB<t>BDSB. 



Adding similar results for all such columns and passing to the 

 limit we obtain the first of the relations given above. The re- 

 maining two are obtained by employing columns parallel to OY' 

 and to OZ'. In the derivation of [368] by means of this the- 

 orem the function 4> is Xx'^x. 



Suppose, however, that in the above proof (i>{x', y', z') is dis- 

 continuous at a certain surface s" which divides the region of 

 integration into two parts, li AB (Fig. 8) intersects this sur- 

 face s" in C then as we approach C in passing along BA from B 

 the function <f>{x', y', z') reaches as a limit a value </>ci which 

 differs finitely from the limit </)c2 which is reached as we ap- 

 proach C along AB from A. In applying Green's theorem now 

 we must apply it separately to the two regions and integrate 

 (d4>/dx') dx' dy' dz' first along a column stretching from B to 

 C taking 0ci as the value at C, and then along the column 

 from C to ^ taking 0^2 as the value at C. In this way we ar- 

 rive at the result 



—f dx' dy' dz' (throughout the column) 



= as' 4>B Dsb' + aci" <i>ci DSc" + otc-l' 0c2 -DSc" + ola! <^a Ds/, 



where the direction cosines with the suffix 1 are for the normal 

 to Dsc" directed outwards from the first part into which the 



