Problem of Three Bodies. 1 3 



,x"d*x" + y"d*y"+z"d?z" , XTT3 rt - H .. 



+ 7ww' ,^ hNR = 0. If we combine 



at 1 



this with the equation immediately preceding (M.), we have 



4- mm' 



ai ~ r~- • • (N.) 

 „ XT /Mm Mm' wm'\ „. 





If § be the distance of the centre of gravity of m and m' from 



M, we have mr 2 + m' i n sa ft," g 2 H y- r" 2 . By this elimi- 



nating d 2 (r 2 ) from (N.), there results 

 „ d 2 (g 2 ) Nmw' <? 2 (r" 2 ) 



d* 2 + a" d* 2 



M/*" 



xt /Mm 



F 



+ 



Mm' 

 r' 



+ 



mm'\ 



2 b. 



(O.) 



Let P = — + 

 r 



dP 



ji> 



its properties are 



dP 



dP 



c?v d# rfr ^ 



«?*/' 





= o, 



<*P f*P ^/^P_ ;rfP 



e?# rf<2? c?^ da' 



0, 



dP dP , ,tfP ,rfP 



v dz dy " dz' dy' 



Equations (A.) will be replaced by 



d 2 x imx _ ,dP d*V , py _ ,dP d 2 % fiz _ , dP 

 Jp + 73 - m jp -Jp + -7a - m jjn JJ2+ 73 ~ m -J3 ; 



dV tl^__ dP d^_ M dP d?J f£* _ d_P 



dt 2 + P 5 ~ m dx> dt 2 + r' 3 ~ m dy 9 dt 2 + r' 3 ~~ m dz' 

 From these, by the aid of the foregoing properties of P, we 

 without difficulty find 



x dd 3 — y d d 2 + z d d x = 0, x 1 d c 3 — y 1 d c 2 + z' d c x = 0. 

 These we have before found in a different manner. They are 

 only introduced here to give an example of the use of differ- 

 ent perturbating functions. 



On some occasions perhaps the following form may be used 

 1 ™i 



m 



with advantage : V = — T 



ITT'" 



d 2 x fix __ d V 



dx» 



d^y A 6 y _ d V d 2 z fJbz _ dV 



