determining the Orbits of Comets, 9 



. . , cos \ (1/ — v) cos fVrr 1 



1 + tan \ t/ tan \ v = M r 2 - = — , 



2 2 cos J if cos \ v q 



and likewise for the one factor its value, viz. 



. . . sinj(i/ — t;) sin/ V rr* 



tan id — tan A a = fj r~ = — * > 



z "* cos £ 1/ cos \ v q 



we obtain 



(6) F = K* V2 ?= 2rV sin/cos/ + | ^ 7 sin/ 3 , 

 or after substitution from (4) and (5), 



KtV2q = ±b(m + n)mn\f2q + % v ^ ' V q 9 



therefore 



2K/=+(/»+«)»!» + j(»i + n) 3 = 3 (m 3 + n 3 ), 

 or 



(7) (r + T 1 + k)*+ (r + r* — k)* = 6Kt. 



This equation is solved by trials in Olbers's method. Ap- 

 proximate values of k, r 9 r' are substituted until the equation 

 is satisfied. As it is evidently quite indifferent what form is 

 given to it, we may just as well deduce the value of one of 

 the variable quantities from it. 



The powers being expanded in series, the odd terms will 

 destroy one another, (having for the moment only regard to 

 the upper sign,) and we shall have 



6K/=3^(r + r')---i^F(r + r')---i||^F(r-fr')"'"&c. 



or 



2K* 



k 1 / k \ 3 1.3.5 / k y 



(r + r')*~r + r ! 4,6\r+^/ 4.6.8.10\r + r7 °* 



k 

 Reversing this series, so that the value of :i is de- 

 termined from it, and putting for brevity 



(8) - 3 = >), we shall find 



V (r + 7 J )* 



r + i* ' ? 24 ' ' 384 ' 9216 



or 



* Lambert's Insigniores Orbitcc Cometarum Proprietates, p. 63. 

 Third Series. Vol. 7. No. 37. July 1835. C 



