CASE VII. 



103 



At the point A, fig. 13, we have also 



Kdfjw) ^ (/c +l)m^ ^ 



2 dw ^'^(M-wy 



which becomes by virtue of (71) and (66) 



--H[M-(/c + l)wP=0 

 u 



agreeing with (61), the condition for a maximum or minimum (or minimax). 



Fig. 14. 



The point B of fig. 14 corresponds to the point B of fig. 13. At the 

 point B, fig. 13, we have 



Aw)=0 



dfiw) 

 dw 



= 



which lead again to equation (61). Therefore the smaller real root of (61) 

 belongs to a minimum and the larger to a minimax. 



However the system may change under loss of energy the w and v of 

 fig. 14 must always go from a curve of higher to one of lower energy. When 

 the energy is greater than that for which the point A appears in fig. 14, 

 the curves give us no positive knowledge regarding the series of changes 

 the system may undergo, except that if at any time w = u + v<M it has 

 always been, and will always be, less than M, and the opposite. But if 

 the energy is less than that for which A appears in fig. 14, then, if at any 



