CASE VII. 101 



which, after extracting the cube root, is 



M = (^^)' + (l+«)w 



Let the common value of 7), and Dj be D. Then this equation becomes 



M =Di + (1 +«)^^ =Z)i +[m, + (^) m,]^ (62) 



But when Di = D^ = D, (55) becomes 



M=Pi+K + (^^)m,]l (63) 



where P* must be taken with the positive sign. Since (62) and (63) must 

 simultaneously be satisfied for a maximum or minimum, we must have, 

 when E is a maximum or minimum, 



P = D, = D, (64) 



That is, the energy is at a maximum, or minimum only if P= +D^= -\- D^, 

 i.e., if the whole system moves as a rigid body. 



There can be a maximum or minimum only if (62) has real roots. The 

 treatment of this question is the same as that given in IV, except that we 



must replace Wj of that section by Wj + f— ) wig. Hence the condition for 



real roots, and therefore for a maximum and a minimum of E, is, by (31), 



M ^ i- [m, + (^)' m J =|-(1 + «)*m,J (65) 



Let us suppose the inequality (65) is satisfied and then consider the 

 surface defined by (58). It will be most convenient to give E a series of 

 constant values and to draw the corresponding equi-energy curves. To 

 simplify the treatment let 



w = u + v (66) 



Eliminating u from (58) by means of this equation and solving for v we find 



l±i„„„±J_!£!+li+^^+li±^!>^ (67) 



K \ K k{M — W)^ KK 



Consider first the function 



/(^,)==_^ + ll+^^- + li±^ (68) 



whence 



''-^v=w±^f{w) (69) 



