._-.,; BILLS AND DALES. 



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, 



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 / is the number of 

 Bottoms or Immits and B the number of Bars, then 



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

 reckon a pass as single, double, or n-ple, according as two, three, or n + 1 

 regions of elevation meet at that point, and a bar as single, double, or ?-ple, 

 aa two, three, or n+1 regions 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 them ; but if we consider 

 the regions cut off by the curves of constant value of the function, we may 

 call p 1 of them false maxima and q 1 of them false minima. 



On Functions of Three Variables. 



If we suppose the three variables to be the three co-ordinates of a point, 

 and the regions where the function is greater or less than a given value to be 

 called the positive and the negative regions, then, as the given value increases, 

 for every negative region formed there will be a minimum, and the positive 

 region will have an increase of its periphraxy. For every junction of two dif- 

 ferent negative regions there will be a false minimum, and the positive region 



