8 Prof. Encke on Olbers's Method of 



we have, by integration, 



(1) K/</2? = 2? 9 (tani*/-tan£u + itani*/ 3 -£tani*; s ). 



Introducing instead of the true anomaly and the smallest 

 distance, the radii vectores and the chord between, we obtain 

 the important advantage of exterminating the elements of the 

 parabola from the equation. In applying the method of co- 

 ordinates it is likewise more convenient to retain distances 

 only in the formula. With this view put 



(2) if^inm <2f t 



and taking k for the chord, we have 



^ = r 2 tr' 2 -2rr' cos 2/ or = (r + ? J )* — 4 r 1 J cos/ 2 ; 

 consequently 



2 cos/ */ri J = ± s/ (r + r* + k) (r + r' — k). 

 For the sake of brevity we make 



. , r + r' + k = m 2 



{ ' r + r l -k= n\ 



which is permitted in every case, even with the condition that 

 m and n shall be always positive ; and we have 



r + r'=J(m*+»*) 

 2 cosy s/ r f* = + m n. 



The upper sign for which, f /_ 90°, i/-»Z 180°, may be 

 almost considered as the only one here to be referred to. 



By the same quantities sin/ may likewise be expressed. For 

 we have sin/ 2 = sin \ (t/ — vf 



= cos \ v* -f cos \ t/ 2 — 2 cos \ (i/ — b) cos J v cos J i/ ; 

 and by introducing r and H we obtain 



™ Q Q cos f 



J r r 1 l \Srr' 



r -f- ? J — 2 cosy >/ r r' 



and by substitution from (4), 



(5) 2 sin/Vr / = O + ») -/ 2 ?, 



where, as above, the upper sign is to be taken for/^ 90°, and 

 the lower one for/ 7 90°. 



Separating in (1) the factor contained in the part on the 

 right-hand side we have, 



F =zKt s/1q = 2 £ 2 (tan \ d — tan \ v) 



(1 + i tan \ t/ 2 + J tan £ t/ tan £ + £ tan i v 2 ) ; 

 and placing in this equation instead of 1, its value deduced 

 from this equation, 



