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



them ; but if we consider the regions cut off by the curves of 

 constant value of the function, we may call^> — 1 of them false 

 maxima and g — 1 of them false minima. 



On Functions of Three Variables. 



If we suppose the three variables to be the three coordinates 

 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 different negative regions there will be a false minimum, 

 and the positive region will have a diminution of its periphraxy. 

 Hence if there are q true minima there will be g — 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 mi- 

 nima and the false ones. 



In like manner, when a negative region expands round a hol- 

 low 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 arejo maxima there are/? — 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 sta- 

 tionary 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 



