Plummer — Note on the Use of Conjugate Functions. 11 



Also it is easy to show that 



d fdTA d'l\ 



, = - -InJij. 



dt \ d %j a§ 



The equations of motion thus become 



|(4)-2^4|{ W+ F-^)} 



If then we put dt = Jdr, these become simply 



d% T d n dV d\ dl; dV 



— - 2vJ — = -^- , y-, + 2nJ — = — — , 



where V = J {V + ±n-(o? + if) - h], 



is the equation of energy. 



6. If we write /, = /(? + i v ), f, = /(£ - ir,), 



then r 2 = a; 2 + f = /,/,, and J" = fj\. 



Let us consider the case of central forces, n = 0, P" == fir p *\ If 

 we put 



» + ty = /(£ + «?) = (? + *i)*> 



then F' = ft 2 (? + rff- 1 |/t(? + „ 2 )**c +1 > - A}. 



The first term becomes constant, and the second corresponds to a central 



force varying as p q , if 



ft - 1 + p (p + 1) = 0, 2 (ft - 1) - g_ + 1, 



q + 1 



whence (a- + 3) (p + 3) = 4, ft = - 5 • 



v 7 ^ _p + 1 



The different eases are thus associated in pairs, except that of the force 

 '/•" 5 , which corresponds to itself, and that of r' 1 , which is otherwise excluded. 

 The problems soluble by elliptic functions are arranged in pairs thus : — 



p = 5 3 -\ -4-5. 



o 



k - 4 3 3 4 ~ 2 _1 - 



5 7 5 3 



2 = '2 "3 "3 "2 ' 7 " 5 - 



It is natural that any one of these problems is converted into one of the 

 same class ; and it is also to be noticed that the relation between p and q 



