60 



DEFINITE INTEGRALS. 



[INT. III. 



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 denned 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. 15, is quad- 

 ruply-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 dia- 

 phragms, 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 deri- 

 vative dW/dx 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 dW/dx in the space T multiplied by the volume r. 

 We have at once 



FIG. 15. 



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



respect to x, the volume is divided into elementary prisms whose 



Y 



FIG. 16. 



sides are parallel to the .XT-axis, and whose bases are rectangles with 

 sides dy, dz. 



