761 



differs from my former method in not nsing the symmetry of the 

 field for decreasing the number of functions to be calculated. This 

 would scarcely be an advantage here, the functions having all the 

 same to be calculated from partial differential equations. 



1 have worked out the calculation on the supposition that the 

 bodies have invariable shape and volume, and I have carried the 

 calculation only to such an extent as is required by the precision 

 with which we are going to calculate the equations of motion of a 

 particle in it. As to this precision it will be sufficient that it furnishes 

 a first correction on the equations of motion as defined by classical 

 mechanics and Newton's law. 



^ 1. We call ^f^\ y^"^ c the values of the (juantities//^, y,.,, '^ — //, 

 as they would be if both centres were absent. If only the first centre 

 existed, these quantities would \'>^ jf^') -\- g']!}- y^^^-j-y',' , c-)- f'*'; ^vhen 

 we have only the second centre we represent them by y{'>-\-y^'\ 

 y('j) _|- yC.2)j c-|-c(2), and in the case of both centres we put: 



^„=.<,W+^(i)+<,(2)+"^,,, y,,=y(Ol + y2) + y(2)4 y,,, \/rg=c i^c'^^^ c'^^+'e (1) 

 The quantities g'p, -f^'K c are constants. We know already 

 g''^)^ y(i), c^i), gi'^), y(2), cf-'^) -^ they do not contain terras of order zero. 

 (We name one the order of a term such as X/r or v-/c'^). The 

 quantities g^j, y,v, (' finally do not contain any terms of lower order 

 than the second; it is they that must be calculated. 



We now have to substitute the expressions fl) in the equations 



2 ^(l^^9U^ih.^^)-=-''-{^^^+U.) .... (2) 



Omitting all terms of higher order than the second, we have 



dy,. 



l/-i/y.,3(/.^^— = 



(c + c(i)+c(2))(y.,3(o)+y.,3f i)+y.,,/2))(r7,.(0!+^.,/.i)+,^,,/2))f %^^ + %^ + p^^ 



\ Ox^ dx^ ox^ ) 



■cyx,2^"^(7j 



W(7c,;o) 



0*',3 OA',3 



Ox^ 

 49* 



