426 Mr. J. C. Maxwell on Hills and Dales. 



though they do not actually join them ; but when the streams 

 increase in quantity, they join and excavate courses for them- 

 selves ; and these actually run into the main watercourse which 

 bounds the district, and so cut out a river-bed, which, whether full 

 or empty, forms a visible mark on the earth's surface. No such 

 action takes place at a watershed, which therefore generally re- 

 mains invisible. 



There is another difficulty in the application of the mathema- 

 tical theory, on account of the principal regions of depression 

 being covered with water, so that very little is kuown about the 

 positions of the singular points from which the lines of water- 

 shed 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 x be the number of single passes, p 2 that of double passes, 

 and so on. Let b lf Z> 2 , &c. be the numbers of single, double, 

 &c. bars. Then the number of summits will be, by what we 

 have proved, $ = l+ Pl + 2p 2 + &c, 



and the number of bottoms will be 



I=±l + &i+£ft s +fee. 



The number of watersheds will be 



W=2 (*!+/>,) +3(6 2 +^ 2 ) + &c. 



The number of watercourses will be the same. 



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



P-L + F-R=0, 



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 sur- 

 rounding 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. 



If we put L equal to the whole number of lines, and P equal 

 to the whole number of points, we find that F, the number of 

 natural districts named from a hill and a dale together, is equal 

 to W, the number of watersheds or watercourses, or to the whole 

 number of summits, bottoms, passes, and bars diminished by 2. 



