Plummer — Note on the Use of Conjugate Functions. 9 



Finally, in the case of a repulsive force, the equation of energy becomes 

 2 j^J+ (JJj = - m + h(?+ .r) and h = + M /2«. 

 The appropriate solution is now 



I = /(TTl)^ cosh 1Q%, , = y (TTT^ sinh ^)%, 

 which lead to 

 x = ae+ a cosh f - ] t, y = a ^/e % - 1 sinh - r, r = ae cosh (-) t + a. 



. fa 3 \U . ,(u\i />\J ) 

 Hence, < = — {esinh - r + - r), 



which completes the solution for every case. 



4. Bohlin has indicated the application of his transformation to the 

 problem of three bodies. An analogous transformation has been given 

 by Thiele (Astr. Nachr., 3289), which applies with advantage to the 

 restricted problem of three bodies in which the two finite masses are 

 equal. This is 



x + iy = cos {E + iF), 



or x = cos E cosh F, y =« - sin E sinh F. 



This is simply the transformation to elliptic coordinates commonly 

 employed in the problem of two fixed centres of attraction. But further, 

 the time t is changed to ^, where 



dt = ?v 2 c&//, 

 and r lt r 2 are the distances of the third mass from the two finite masses. 

 The equations of motion then take the form 



Jl XT JJjl 



f — - (cosh 2F- cos 2H) ~rr = i sin 4# - \K sin 2E 



d 2 F dE 



— + (cosh 2F - cos 2E) -j- = \ sinh <±F - \K sinh 2F + 8 sinh F, 



where K is a constant occurring in the equation of energy. 



An orbit of special interest, called an orbit of ejection, is that in which the 

 small body is projected from one of the finite masses, and, after describing a 

 relative path resembling a cardioid in shape, returns to the mass from which 

 it started. The result of the above transformation is that the corresponding 

 trajectory in the (E, F) plane is a simple closed curve free from any singularity, 

 performed with a velocity which is everywhere finite. This orbit has been 

 investigated by Burrau and Stromgren (V. J. S. der Astr. Ges., xlviii, p. 222). 



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