Olbers's Method of determining the Orbits of Comets. 125 



acquaintance, and that my letter is written without his know- 

 ledge, except, perhaps, as one of the Editors of your valuable 

 work. I have the honour to be, 



Gentlemen, yours, &c, 

 July 18, 1835. X. Y. Z. 



XV. On Olbers's Method of determining the Orbits of 

 Comets, By Professor Encke. 



[Continued from p. 25.] 



I^HE general differential equations for every celestial body 

 of our system, if x' and y f denote its heliocentric coordi- 

 nates in the plane of its orbit, are 



dt* + r' 3 " U dt* + r' 3 " U 



in which the terms are supposed already to have been multi- 

 plied by the constant K. These equations give the following 

 by further differentiation : 



a?x' _ dt ; J_ dx[ 



dt? " + r' 4 X r' 3 ' dt 



d 4 x' _ fj_ 12 /dr[ V 2 3^ jPrTi i 

 ZF~ + Lr' 6 r' 5 W*/ + r' 4 ' d t* J * 



6 rfrj d.r f 



and similar expressions for ~-*L and —jr* by changing a:' for 



y . If these values are substituted in Taylor's formula, every 



x and y may be expressed by a series in which only the first 



differentials of the coordinates are to be found, and in which 



dx f 

 the coefficients of x' and -j- (which coefficients involve r 1 and 



its differentials together with the powers of the time) in the 

 expression for x are identical with the coefficients of y' and 



-2- in the expression for y. 



Putting, for brevity, 



K (*' - t) = t" 

 (25) K (t" - f) \m t 



K (I* - /) = if 



and likewise, 



R 2 



