determining the Orbits of Comets. 129 



The quantities <r and cr" are independent of the intervals ; 

 but 2—2' and 2' — 2 n and consequently the numerators and 

 denominators of M' and M" will generally be of the first order 

 of the intervals, and their values will not be liable to greater 

 errors than the observed a! — a and V — 8. For we have 

 sin <r sin o 7 sin {2 — X) 



= sin (&' - 8) sin (£(« + «')- 6') cos £ (a' - a) 

 - sin I (a - a) cos Q (a + a') - 6') sin (* + 8). 



If, however, in a particular case, 2 — 2' and 2' — 2" should 

 be so small that they might be considered as quantities of a 

 higher order, or should they be = 0, the expressions for M' 

 and M" and consequently M will be either too unsafe for 

 calculation or perfectly indeterminate, and the possible errors 

 of observation as well as the term neglected in M might obtain 

 such a preponderating influence that the use of an expression, 

 otherwise very approximate, might lead to essential errors. 

 At least the accuracy of the approximate expression of M, the 

 neglected factor M" being now of the order — 1, would be of 

 an order less high by one. 



This case of exception will, however, as the view of the for- 

 mulae proves, only take place when the three geocentric places 

 of the comet lie apparently in a great circle with the middle 

 place of the sun, or too nearly approach to such a position. 

 It would, however, be wrong to consider this as a case of ex- 

 ception to Olbers's method, as it is founded in the nature of 

 the problem and would and must take place in like manner, 

 in every method (not excepting that of Laplace). Every me- 

 thod which makes use of the important condition of a motion 

 in a plane passing through the sun is liable to it, and must be 

 so, although it is not always mentioned ; and every method 

 which neglects this important and simple condition and yet ap- 

 plies no more data of observation, making consequently the 

 problem perfectly determinate, will on that account make a 

 sacrifice in point of accuracy. If we leave for a moment the 

 motion of the earth out of the question and put 6=6'= 6", 

 the numerator and denominator of M are factors of the pro- 

 jections of [r r'] and \r r"] on a plane which is perpendicular 

 to R'. If these become = 0, or too small, the conclusions 

 which may otherwise be derived from their ratio to one another, 

 joined with the condition of a constant plane of motion, as to 

 the distances of the comet from the earth, loses entirely or 

 partly its force, and one is obliged to introduce another, al- 

 though hitherto unknown quantity, perhaps the distance of 

 the comet from the sun, in order to determine the ratio of 

 the distances with equal accuracy. This is indeed the man- 

 Third Series. Vol.7. No. 38. Am. 1835. S 



