300 PROCEEDINGS OF SECTION A. 



then coming back to B, then coming back to A, and so on, so that 

 any infinitely small element of the portion of the strip bet\A'een A 

 and B (portion shaded on the figure) is gone over n times in one 

 direction and n times in the opposite direction, whilst every 

 infinitely small element of the portion of strip inside A is gone 

 over n times in one direction and {n — 1) times in the opposite 

 direction."^' Therefore, dividing the surface into slices by planes 

 parallel to xoz, we see that, when we go once over the surface, we 

 go n times in one direction and n times in the opposite direction over 

 every element of area between A and B, whilst we go n times in 

 one direction and n — 1 times in the opposite direction over every 

 element of area bounded by A. 

 But as obviously 



/fc=-//-c 



the index indicating that the integral is taken over any element of 

 area C between A and B in one direction and the index — C that 

 the integral is taken over the same element of area C in opposite 

 direction (the same single-valued function being under sign fj 

 we can replace the dovible integral of (-) by 



//'sc _ :sc + A 



that is to say the integral of the same function over elements of C in 

 one direction, over the same elements in the opj^osite direction, and 

 over A in the standard direction withoiit changing anything. 

 Applying, then, the same transformation as in the first case 



./T'sC - EC + A 

 becomes, by an appropriate choice of 2C, an integral all over 

 surface, z = ffx, y), and the theorem is still true. 



Now, considering three functions, ?<, v, iv, single-valued and 

 continuous (and therefore finite) of x, y, z, all over a certain surface 

 ffx, y, zj =z we can write — 



where /, m, n, are the direction cosines of normal to surface 



reckoned in positive direction. And making the sum we obtain — 



I ^9« _ ._.\ /^, _ ^A J9V _ 9«\ 1 ^g 



•'•' ( \9y dz) ^ \dz, dxj \dx dyj \ 



■=■ fudx -}- vdy •\- 10 dz 



' It is clear that accoi-ding to the shape of the surface z=f{x, y) n may vary from 

 one element to another. 



