Plummer — Note on the Use of Conjugate Functions. 13 



(x, y) in this term is the centre of gravity of the large masses, and not the 

 point midway between them. But if ^ = p 2 = p, we have 



V = 2fic cosh t\ - \&h (cosh 2rj - cos 25) + tV^ 4 (cosh 4rj - cos 4?) 





1 fdP\* 



- m) + 



l(dr,\\ 

 2\dr) ' 





and accordingly, 







d?' 



- w? (cosh 2rj 



- cos 21) 



dt) 



dr 



= - c-h sin 2£ + ^nV sin4£, 



.72 Jp 



-=-„ + nc 2 (cosh 2rj - cos 2£) — = 2uc sinh r? - cVi sinh 2i ( + |m, 2 c 4 sinh 4n. 

 «r 2 ar 



If in these equations we put n = 1, c = 1, 2p = 8yiV, or p. = 4, 7i = fiT, 



S = i?, ») = -f, and r = i//, we have at once the equations of § 4. Here the 



dependent variables are not separated, and it is seen immediately how the 



simplest case of the problem of three bodies transcends in complexity the 



problem of two fixed centres. 



9. A purely algebraic transformation may be worth noticing. This is 

 x + iy = \ c { (£ + i v y + ({= + ^)" 2 }, 

 and it is convenient to write 



£ = p cos 0, r\ = p sin 0. 

 Then J = ///,' = c>- 2 (p 4 + p" 4 - 2 cos 40), 



* = |c (p 2 + p~ 2 ) cos 2<p, y = \c (p 2 - jtT 2 ) sin 2ij>, 

 r i = h c (p 2 + p~ 2 + 2 cos 20), r 2 = \c (p 2 + p~ z - 2 cos 20), 



V= JL + JL = ^ (<■>' + P"> 



»*i n e (p 4 + p -4 - 2 cos 40) - 



|?i 2 (* 2 + y 2 ) = in 2 c' (p 4 + p" 4 + 2 cos 40). 

 Hence 



F = 4pc (1 + p" 4 ) + i %Vp- 2 (p 8 + p- e - 2 cos 80) - e 2 V 2 (p 4 + p- 4 - 2 cos 40) 



which can be easily expressed in terms of % and r\. The expressions 



dV dV" sin0 dV 



^ = coB ^--p- w 



dV . dV cos0 dV 



-—— = sin — — + — — , 



or\ dp p 0<f> 



are probably too complicated for any practical use. The effect of the 



transformation is, however, to give the equations of motion in a form which 



involves only rational algebraic functions of the variables. The pole p = is 



not of consequence, since it corresponds to a point at infinity in the (x, y) 



plane. 



