Of 



20 Prof. Encke on Olbers's Method of 



d g = -j- du dg' = -j- du 



d r = sec 8 cos C d p 



dr"=sech"cosC"dg". 



If we designate the value of s originally assumed by s , its 

 correction by d s , so that 



^0 ~f~ ^ ^0 == ^9 



or the true value again by s p the value of s, resulting from the 

 assumed value s , and its correction by ds l9 so that in like 

 manner s x + ds l = s = the true value, we have 



f p sec 8 cos C + p" s ec S" cos C" ^ F , 

 dSl ~ m 1 2 (r + r") J ' 7*7*'' 



in which formula the angles C and QI' are those angles which 

 represent the angular distance of the earth from the sun when 

 seen from the comet, or the angles C and C /r , in the triangles 

 S C E and S C" E". The formula supposes that the assumed 

 value 5 is not so far distant from the truth as that the differ- 

 ential for s may not be applied with some approximation ; 

 it is likewise clear, that by d s 09 ds iy may be understood not 

 only the corrections of the numerical values but likewise the 

 corrections of their logarithms. 



On account of the circumstance just mentioned, we may 

 simplify the formula by putting for the sum of the terms 

 g sec 8 cos C and g n sec 5" cos C", this one, 2 g 1 sec 8' cos C, by 

 which, at least in reference to the general form, nothing is 

 changed. We may besides designate for the sake of simplicity 

 p 1 sec 8', or the true distance, by the single letter J'. Besides, in 

 order to have such quantities only as have a geometrical mean- 

 ing, let us consider the triangle N C C" above referred to. The 

 sides of this triangle are k, g h, g, and the angle opposite to 

 k is $. Let the angles opposite to p h and g be denoted by % 

 and x "; we have § h , u = k cos x " • ph 



= k(k -gcosx). 



By this means we obtain this simple form, if 

 _ A 1 cos O k 



(r + r") ' h — g cos x 

 ds { = — q . ds . 



The factor q can only become negative when either cos C 

 becomes negative, or g cos % V h. The first case supposes 

 that the comet is within a sphere, the centre of which is the 

 sun, and the radius the distance of the earth from the sun, so 

 that it would be very near both to the sun and to the earth. 

 The second case, on the contrary, requires that g 7 £, or as 



