16 Prof. Encke on Olbers's Method of 



M _ t" — t' tan 8' sin (a — ©') — tan 8 sin (a 1 — ©') 



(16) t' - t 'tan 8" sin (a'-®') - tan 8' sin(a"-®') 



p» = M P , 



and have consequently an expression for ■*— in given quanti- 

 ties only. 



The formula (15) contains the only approximate supposi- 

 tions made in Olbers's method. Their admissibility and the 

 cases of exception will be investigated afterwards. 



It is now required to find, by means of the equations (16), 

 convenient expressions for r, r, and the corresponding chord 

 k 9 which shall contain no other variable but p. For the first 

 two there is no difficulty, as 



r 2 = x* + y* + s 2 



The expression for k, for which we have 



F = (*" - *)* + (y" - yf + (z'i - z)% 



is more easily found by a simple geometrical construction which 

 we shall again apply hereafter. Let the place of the sun 

 in space be denoted by S, the two places of the earth at the 

 first and third observations by E and E", those of the comet 

 by C and C". Let a line be drawn though E" parallel to, on 

 the same side with, and equally long as, E C. Calling the ter- 

 mination of the line thus drawn N, and designating its helio- 

 centric coordinates by x i9 y t9 z t9 we have, by what precedes and 

 by the construction, 



x = p cos a — R cos 

 y . = p sin a — R sin 

 z = p tan 8 

 x" — Mp cos a" — R" cos ©" 



(17) y = M p sin a" — R" sin ©" 

 «"sMp tan 8". 



x { — p cos a — R" cos ©" 

 yj ss p sin a — R" sin ©" 

 z t = p tan 8. 



The two first of these three systems give, by adding the 

 squares, 



( . r 4 = p 2 sec8 2 -2 fJ Rcos(a~©) + R 2 



IW fin m M 2 g » sec |ffs -2M p R" cos (*" - ©") + R" 2 . 



Combining the first system with the third, and assuming 

 . v .r — *, = R" cos ©" — R cos = g cos G 

 * ' y-y,= R"sin©"- R sine =g sin G, 



