(io) *=i+^ + ^ 4 + ^>i e 



10 Prof. Encke on Olbers's Method of 



If we consider the intervals of time as small quantities of 

 the first order, ij will be of the same order. The factor of 



2 K* 



* // , jv will, consequently, differ from unity only by quan- 

 tities of the second and higher orders, and may be conveniently 

 represented in a table with the argument >j. The calculation 

 is then very simple. The approximate value of r, r 1 , k being 



2 Kt 

 given, -J-J-. ZT\ * s ^ rst calculated, from which >j is imme- 

 diately derived, and this gives by the table immediately the 

 second factor. The comparison of the value of k thus found, 

 with the one assumed, will prove how far that value is correct. 

 The value of this factor, which I will designate by ft, so 

 that 



I « 5 4 gg 



24 ^ + 384^ 9216 

 may also be expressed by a finite formula, by which the calcu- 

 lation of the table is facilitated, and its use immediately ex- 

 tended to all cases. 



k 

 We may always put — — — , = sin y, whence 



G K t 



- — r = +siny)^ + (l -siny)^. 



(r + r 1 )* 



Assuming, what we may assume, that y always Z 90°, we 

 may use this formula, 



(cos i y ± sin J y) 2 = 1 + sin y, 



and the above equation will be transformed thus, 



6 K t 



(cos \ y + sin \ y) 3 + (cos \ y — sin \ y) 3 = — . 



(r+ i*Y 

 Taking first the upper sign, we obtain 



6 cos ^y 2 sin ^y + 2sinly 3 = - — , 



(r + r) T 

 to which this form may be given, 



/ sin J y \ _ / sin j y \ 3 _ 6Kt 

 \ V2 J \ V2 ) &(r + r>)v 



Lambert's formula shows at once, as h Z. (r + r 1 ), that in 

 both cases also 



Making, therefore, 



6K/ . i 



— : = sin 0, 



2'(r + r / )* 



