32, 33] 



DEFINITE INTEGRALS. 



59 



FIG. 13. 



vector parameter of a scalar point-function will be called a laminar, 

 or lamellar vector (Maxwell). 



The scalar function c/> (or its negative) will sometimes be termed 

 the potential of the vector R. 



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



have supposed in 31 that it was 

 possible to change the path 1 from A 

 to B into the path 2 by continuous de- 

 formation, 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 7, Fig. 13, 

 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. 13 between the contour C and the island 7 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. 



A three-dimensional space bounded externally by a single 

 closed surface is not made doubly-connected by 

 containing an inner closed boundary. For instance, 

 the space lying between two concentric spheres 

 allows all paths between two given points to be 

 deformed into each other, avoiding the inner sphere. 

 But the space bounded by an endless tubular surface, 

 Fig. 14, is doubly-connected, because we may go from 

 A to B in either direction of the tube, and the two 

 paths cannot be deformed into each other. We may 

 make the space singly-connected by the insertion of 



FIG. 14. 



