114, 115] INFINITESIMAL SPACE -ELEMENTS. 



Cylindrical coordinates, 0, 0, co, 



339 



13) 



dS z = yd yd a, element of area of plane, 

 dSg = Qd&dz, element of area of cylinder, 



(a = dQds , element of area of meridian plane, 



dr 



Elliptic coordinates, A, ^, v. 



dpdv >/ Q -v}(\i- 1} (v - 



2 + *0 (& 1 + fQ (c 2 + p) ( 2 + *) (fc 2 + r) (c 2 + v) 

 d <?X ]/(v V) (v ft) (JL a) (1 y) 



14) 



ellipsoid, 



hyperboloid, 



hyperboloid, 



115. Connectivity of Space. Green's Theorem. We 



supposed in 30 that it was possible to change the path 1 from A 

 to B into the path 2 by continuous deformation, without passing 

 out of the space considered. A portion of space in which any path 

 between two points may be thus changed 

 into any other between the same two 

 points is said to be singly -connected. 

 For instance, in the case of a two- 

 dimensional space, any area bounded by 

 a single closed contour will have this 

 property. If, however, we consider an 

 area bounded externally by a closed 

 contour 0, and internally by one or more 

 closed contours J, Fig. 119, such as the 

 surface of a lake containing islands, it 

 will be possible to go from any point A to any other point B by 

 two routes which cannot be continuously changed into each other 

 without passing out of the space considered, that is traversing the 

 shaded part. 



The space in Fig. 119 between the contour C and the island / 

 is said to be doubly -connected. We may make it singly -connected 

 by drawing a barrier connecting the island with the contour (7, 

 represented by the dotted line. If no path is allowed which crosses 

 the barrier the space is singly - connected. 



Fig. 119. 



