164 PROCEEDINGS OF THE LMERICAN ACADEMY. 



If w and v are identically equal, the last equation becomes 



///''/. v -dxdydz — g I J "'[ — -cos (./•,/') + - • cos (j/,/i )d8 



-'///[ey+sw***-^-^" » 



(35) 



(I) If »S' is a closed cylindrical surface the generating lines of which 

 are parallel to the z axis, and if SI, SI' — two functions which within 

 S, satisfy the equations L (Si) = (», L (Si) — — (1) vanish at all points 



W 



TIME. 



Ik. i re 50. 



of 8 and at all points within 8 for which z is positively infinite, ami 

 (2) have the given constant value l.\, at all points in the xy plane 

 within /So ; then if we apply (35) to the difference between SI and SI', 

 using as a field of volume integration the space inside N„on the positive 

 side of the xy plane (Figure 57), we shall learn that in this space $2 and il' 

 must be identically equal. The value of il within S„ is in do way 

 affected by conditions which a physical extension of the function 

 might be required to satisfy outside \. 



(II) If (So is a closed cylindrical surface, the generating lines of 

 which are parallel to the z axis, it IT is a function which within S„ 

 satisfies the equation A(ir) = (», and it' 



(1) ITand dW/dz vanish at all points within and on S n for which 

 z is positively infinite, 



(2) \V has a given constant value (IT,.) at all points on the xy 

 plane within »S,. 



