138 Astronomical and Nautical Collections. 



sible when r' and r"' differ but little from each other, and k" and 

 T are very small. There is, however, no occasion for these 

 awkward abbreviations ; for although the value of ^' cannot be 

 found immediately from Lambert's formula, yet it may be 

 obtained by a few easy trials, since we have three equations, 

 r' = V (R'« - 2 R' cos (A' - a) ^' + sec^ /3' f-) 

 r" == V (R"'* - 2 R'" cos (A " - «'") M^' + sec^ 5" M^ ^'^) 

 k" — \/ (F + G^' + H ^'-) ; in all which the coefficients of ^' 

 are known magnitudes, expressed in numbers ; so that we 

 have only to assume a value for §', and we obtain those of r' r" 

 and k' by the extraction of the square root only. From these 

 we may find without difficulty, by means of the table of the 

 descent towards the sun in a parabola, or by an easy direct 

 reckoning, the time that ought to elapse between the observa- 

 tions, according to the assumed value of §'. This time, com- 

 pared with the observed time, immediately shews whether we 

 ought to increase or diminish the value of ^\ in order to come 

 nearer to the observation. In this manner we approximate 

 rapidly to the truth, and may at last employ a simple interpo- 

 lation. It will seldom be necessary to make more tlian four, 

 or at most five, suppositions ; and the first of them will not 

 require any accuracy of calculation : at least the determination 

 of the true value of ^' from these three equations will always be 

 much more convenient than the solution of a single equation of 

 the 6th degree. 



§42. 



As soon as the value of g' is determined, the determination of 



all the elements readily follows ; for the computation gives us 



at once r, r'\ §', and |"' := M^'. Now if we call the heliocentric 



latitudes, in the first and third observations, >! and x"', we have 



, tano- (3' e' , . ,, tans: (3 ' p" - , 



sm x' =:: —, — —, andsmx = — °,,, ^ . If also we put 



r r ' 



i, s" for the heliocentric elongations of the comet fiom the 



., c J • ' f' sin (A' — a') . ■ „, g" sin (A"— a'") 



earth, we find sm € — ^ ~ , sm e = -2 i^ ' 



?•' cos X r" cos x' 



whence we obtain the two heliocentric longitudes, which may 



