22 Prof, Encke on Olbers's Method of 



Js 2 



which, if necessary, might be employed in an insulated trial. 

 We may employ ds Y or d s 3 according as the one or the other 

 is smaller. 



The latter selection and in general the more or less rapid 

 approximation of the trials depend on the value of g. The 

 first factor, j'cosC 



r + r" 9 



will in almost all cases be a very small fraction. The mag- 

 nitude of the second factor will, on the contrary, mainly de- 

 pend on the value of %. Agreeably to the construction, v 

 is the angle formed by the chords of the orbits of the earth and 

 the comet, if these chords are so placed together that the first 

 place of the earth coincides with the first place of the comet, 

 and the directions are made to correspond to the directions 

 of both bodies. When the motions are parallel, x w hM conse- 

 quently be = 0, and cos% will have its greatest positive value ; 

 but when the motions are in contrary directions, % = 180°, 

 and cos y has its negative maximum. Hence it follows that 

 likewise this factor will in most cases, at the discovery of a 

 comet, be a real fraction, or will at least not differ much from 

 unity, because by the nature of the case comets will always be 

 first seen when their course is rather opposed to that of the 

 earth, and directed towards the earth, than when both move 

 in the same direction in parallel courses. In the latter case 

 they might have been seen before. The ambiguity which 

 seems to exist in this manner of instituting the trials, viz. whe- 

 ther, on account of u 2 = 1c l — A 2 , u is to be taken positive or 

 negative, disappears in practice almost altogether. By the geo- 

 metrical meaning of u it is clear that it can only become nega- 

 tive when the angle y/' is obtuse, consequently g~7 k 9 and, after 

 what precedes, 5 7 4. Cases of this kind are of rare occur- 

 rence, and may, when they occur, be easily decided. 



It appears, therefore, that this way of instituting the trials 

 has greatly the advantage over the usual process, partly on 

 account of the value immediately to be applied of s = 2, partly 

 on account of the methodical manner in which the successive 

 assumptions result from the first without any arbitrary suppo- 

 sitions. Only when u is very small, which quantity for that 

 very reason will then be derived with less accuracy from the 

 combination of k and A, the proceeding may perhaps be more 

 circuitous. The remarks of Dr. Olbers on cases where, in place 

 of s = 2 immediately, a smaller or larger value may be applied, 



