( 105 ) 



and the two intervals of time t, and Tj. The value namely, of the 

 following expression 



where t, = Tj -|- t,, is of the 6''' order with respect to the intervals. 

 I nse the letters C\, G, and (^3 to designate the tnuUipliers of (he 

 second derivatives in tiiis expression and put 



'~ 12 '^~ 12 ' "- 12 • 



Neglecting the terms of the 6"^'^ order we then have for an arhi- 

 trarj^ function of the time, the relation 



r,F,-C\F,-r,F,-C,F,-^r,F,-C\F\ = Q . . {IV) 



provided this function and its first four derivatives be continuous 

 and finite within the interval t^. 



Applying this formula to the heliocentric distance r and to r% I 

 obtain approximate expressions for the semi-parameter/;, and the semi- 

 axis major a of the elliptic orbit. By eliminating /^ from the two well- 



2 » 1 



known differential equations r r^ -{- r =z p and r'' = , I 



r r" a 



find a diiferential equation which may be easily reduced to 



d' ("l l\ .. p-r 



— - (r') = 2 . According to these relations F = 



ar' \r aj y» 



belongs to F—r, and F = 2( j to F = r\ 



If as before I put z for 7,3, the substitution of F =1 r in formula 

 IV yields the following equation to determine p, 



^1^1 — '»',^ + ^3^8 — Ci^i (P — ^'1) — C,^, (P — r,) — C,z, (p — r,) = 

 whence : 



^ (r, + C,z,) r, - (T, - C, z, ) r, + K + C,z,) r, 

 ^ C,Zr-{-C,z,^C,z, ' • 1 ) 



Through the substitution of F-==.r^ in IV I obtain the equation 



r,^^-r,r,H r3V- 2cYi - i V2C/I - i')-2C3r-i - i) = 



V /', a J \ r„ a J \ r, a J 



whence 



- r,r,^ + T,r,^ - r,r,^ -f 2 f ^ + ^ + ^ 



i = ^^- ^'- ^-- . . (VI) 



2(C, + C, + C3) ^ ^ 



