340 



VIII. NEWTONIAN POTENTIAL FUNCTION. 



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 con- 

 centric 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. 120, 

 Fig. 120. 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 a barrier in the shape of a diaphragm, closing the 



tube so that one of the paths is 

 inadmissible. The connectivity of a 

 portion of space is defined as one more 

 than the least number of barriers or 

 diaphragms necessary to make it singly 

 connected. Thus the space in a closed 

 vase with three hollow handles, Fig. 121, 

 is quadruply - connected. We shall always 

 suppose the spaces with which we deal, in 

 this 'book to be singly -connected, or to be 

 made so by the insertion of diaphragms, 

 unless the contrary is expressly stated. 

 Suppose that W is a point -function which, together with its 

 derivative in any direction, is uniform and continuous in a certain 

 portion of space r bounded by a closed surface S. Then its 



derivative -~ is finite in the whole region, and if we multiply it 



by the element of volume dr and integrate throughout the volume T, 

 the integral is finite, being less than the maximum value attained 



by in the space T multiplied by the volume r. We have at once 



Fig. 121. 



fff % 



If, keeping y and z constant, we perform the integration with respect 

 to x, the volume is divided into elementary prisms whose sides are 

 parallel to the X- axis, and whose bases are rectangles with sides dy, dz. 



The portion of the integral due to one such prism is 



dx. 



