118 LOVETT— PROBLEM OF THREE BODIES. [April 23, 



and if in particular the symbol * indicates the square we have 



(46) 



ri^qi = Si(niyi zp .ri V i — «i^) ) 



from which we conclude that the orbits are represented by the equa- 

 tions 



(47) ;..".g-e.JI^HP. _ y^^ (,• _ I^ 2, 3). 



(d) The case in which each of the functions (f>i and ij/i contains 

 both variables Ti and Zi leads to a multitude of problems in which 

 these functions are subject to the single conditions 



(48) r^ [2(f>i{l - i/r; J + r^cf,.^^ + ^,(/), J - 2i|r../r .^^ = o. 



where flj an arbitrary constant, wi an arbitrary function, the inte- 

 gration follows a course parallel to that pursued under (b) above,, 

 and leads to complicated transcendental equations for the determina- 

 tion of the corresponding orbits. 

 Vienna, 



February 25, 1909. 



(49) 



