164 PROCEEDINGS OF THE AMERICAN ACADEMY. 



If IV and V are identically equal, the last equation becomes 



///"■ • ^ ^'"^ ■ ^"'^^^^ = ^11" (S 



nv 



cos (x, w) + -o- • COS (jj,n) \dS 



(I) If >So is a closed cylindrical surface the generating lines of which 

 are parallel to the z axis, and if 12, rJ' — two functions which within 

 So satisfy the equations L (n) = 0, L (fi') = — (1) vanish at all points 



TIME. 



FiGDKE 66. 



of So and at all points within S^ for which z is positively infinite, and 

 (2) have the given constant value Qo at all points in the .ri/ plane 

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

 using as a field of volume integration the space inside S^on the positive 

 side of the xi/ plane (Figure 57), we shall learn that in this space fi and O' 

 must be identically equal. The value of fi within So is in no way 

 affected by conditions which a physical extension of the function 

 might be required to satisfy outside >So. 



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

 which are parallel to the z axis, if W is a function which within So 

 satisfies the equation L ( W) = 0, and if 



(1) ^Fand dW/dz vanish at all points within and on So for which 

 z is positively infinite, 



(2) W has a given constant value (Wo) at all points on the ,ri/ 

 plane within So- 



