HILLS AND DALES. 237 



will have a diminution of its periphraxy. Hence if there are q true minima 

 there will be q 1 false minima. 



There are different orders of these stationary points according to the number 

 of regions which meet in them. The first order is when two negative regions 

 meet surrounded by a positive region, the second order when three negative 

 regions meet, and so on. Points of the second order count for two, those of 

 the third for three, and so on, in this relation between the true minima and 

 the false ones. 



In like manner, when a negative region expands round a hollow part and 

 at last surrounds it, thus cutting off a new positive region, the negative region 

 acquires periphraxy, a new positive region is formed, and at the point of contact 

 there is a false maximum. 



When any positive region is reduced to a point and vanishes, the negative 

 region loses periphraxy and there is a true maximum. Hence if there are p 

 maxima there are p 1 false maxima. 



But these are not the only forms of stationary points ; for a negative 

 region may thrust out arms which may meet in a stationary point. The negative 

 and the positive region both become cyclic. Again, a cyclic region may close 

 in so as to become acyclic, forming another kind of stationary point where the 

 ring first fills up. If there are r points at which cyclosis is gained and r' 

 points at which it is lost, then we know that 



r = r' ; 



but we cannot determine any relation between the number of these points and 

 that of either the true or the false maxima and minima. 



If the function of three variables is a potential function, the true maxima 

 are points of stable equilibrium, the true minima points of equilibrium unstable 

 in every direction, and at the other stationary points the equilibrium is stable 

 in some directions and unstable in others. 



On Lines of Slope. 



Lines drawn so as to be everywhere at right angles to the contour-lines 

 are called lines of slope. At every point of such a line there is an upward 

 and a downward direction. If we follow the upward direction we shall in 

 general reach a summit, and if we follow the downward direction we shall in 

 general reach a bottom. In particular cases, however, we may reach a pass or 

 a bar. 



