HILLS AND DALES. 239 



earth's surface. No such action takes place at a watershed, which therefore 

 generally remains invisible. 



There is another difficulty in the application of the mathematical theory, 

 on account of the principal regions of depression being covered with water, so 

 that very little is known about the positions of the singular points from which 

 the lines of watershed must be drawn to the summits of hills near the coast. 

 A complete division of the dry land into districts, therefore, requires some 

 knowledge of the form of the bottom of the sea and of lakes. 



On the Number of Natural Districts. 



Let p, be the number of single passes, p^ that of double passes, and so on. 

 Let & & 2 , &c. be the numbers of single, double, &c. bars. Then the number 

 of summits will be, by what we have proved, 



and the number of bottoms will be 



The number of watersheds will be 



W= 2 (b, + Pl ) + 3 (b, + Pi ) + &c. 

 The number of watercourses will be the same. 



Now, to find the number of faces, we have by Listing's rule 



where P is the number of points, L that of lines, F that of faces, and R 

 that of regions, there being in this case no instance of cyclosis or periphraxy. 

 Here R = 2, viz. the earth and the surrounding space ; hence 



F=L-P + 2. 



If we put L equal to the number of watersheds, and P equal to that of 



summits, passes, and bars, then F is the number of Dales, which is evidently 

 equal to the number of bottoms. 



If we put L for the number of watercourses, and P for the number of 



passes, bars, and bottoms, then F is the number of Hills, which is evidently 

 equal to the number of summits. 



