Mr. J. C. Maxwell on Hills and Dales.. 423 



2ndly. A region of depression may thrust out arms, which 

 may meet each other and thus cut off a region of elevation in the 

 midst of the region of depression, which thus becomes a cyclic 

 region, while a new region of elevation is introduced. The con- 

 tour-line through the point of meeting cuts off two regions of 

 elevation from one region of depression, and the point itself is 

 called a Pass. There may be in singular cases passes between 

 more than two regions of elevation. 



3rdly. As the level surface rises, the regions of elevation con- 

 tract and at last are reduced to points. These points are called 

 Summits or Tops. 



Relation between the Number of Summits and Passes. 



At first the whole earth is a region of elevation. For every 

 new region of elevation there is a Pass, and for every region of 

 elevation reduced to a point there is a Summit. And at last 

 the whole surface of the earth is a region of depression. Hence 

 the number of Summits is one more than the number of Passes. 

 If S is the number of Summits and P the number of passes, 



S = P + 1. 



Relation between the Number of Bottoms and Bars. 



For every new region of depression there is a Bottom, and for 

 every diminution of the number of these regions there is a Bar. 

 Hence the number of Bottoms is one more than the number of 

 Bars. If I is the number of Bottoms or Immits and B the 

 number of Bars, then 



I = B + 1. 



From this it is plain that if, in the singular cases of passes 

 and bars, we reckon a pass as single, double, or rc-ple, according 

 as two, three, or « + l regions of elevation meet at that point, 

 and a bar as single, double, or ra-ple, as two, three, or n + 1 re- 

 gions of depression meet at that point, then the census may be 

 taken as before, giving each singular point its proper number. 

 If one region of depression meets another in several places at 

 once, one of these must be taken as a bar and the rest as passes. 



The whole of this theory applies to the case of the maxima 

 and minima of a function of two variables which is everywhere 

 finite, determinate, and continuous. The summits correspond 

 to maxima and the bottoms to minima. If there are p maxima 

 and q minima, there must be p-\-q — 2 cases of stationary values 

 which are neither maxima nor minima. If we regard those 

 points in themselves, we cannot make any distinction among 



