32 
MR. RUMKER’S OBSERVATIONS 
~i" 
A) At 
Then is the reduction to the meridian R = II S — p 2 cotang, z (s — £). 
Demonstration. — Be £ the zenith distance corresponding to the middle time T ; 
the different unknown zenith distances envelopped in the arc 
run through, and z their mean, which is known. Call £ — z' = a, £ — z" 
= b &c. &c Then is 
cos M — cos ? = 2 r sin a l t = r A sin l w 
cos M — cos z' = 2r sin 3 i t —r A 1 sin 1" 
subtract 
cos ? — cos z' = r ( A' — A ) sin 1 " 
„ . !/; r(A'— A sml" r( A 1 — A) z r(A" — A) 0 0 
2 sin i (z — c) = t — — ^ — a = — v o = . &c. &c. 
J v / z + ? \ sin(? + ^<2j sm(? + AO) 
\T _ 2 / 
sin 
2p(A'— A) 
2 a 
_l_ a 2. X / 2 P( A '~ A) , 
ini" ' ’ V cot? sin 1" ^ 
cot? sin 1 " cot ? sin 1 " 1 “ ’ V cot? 
If we now call 2 cot £ sin 1" = <7, we obtain 
_ tan ? /— — ; — -^7 tan ? 
(? + **) 
i _ , i 
cot 9 ? sin 3 1 " a ' cot ? sin 1 " 
sin 
p? \/ 1 + y (A' — A) 
sin P 
or “ = !?-{[> +?(A'- A)]*- l} 
which resolved according to the binomial theorem gives 
(A' - A) _ n% (A' - Af _j_ 3 g 3 ( A f — A ) 3 _ 3.5q*(A'- A ) 4 ^ ] 
_ tan z f v _ „ , _ 9 v ^ ^ , , 
" — sinl"|7 2 2.4 ‘ 2.4.6 2. 4. 6. 8 1“ ~ r J 
Placing now for q its value in the two first parts, and considering that 
A 9 sin l" 
= h according to the construction of Delambre’s Tables, 
" =p (A' - A) - f cot ? (¥ + S - A' A sin 1") + ^ 
tan? 3. 5.q* (A 1 — A ) 4 
sin 1 ". 2 . 4. 6. 8 
And in the same manner 
b - p (A" — A) — p 2 cot £ (&" + l - A" A sin 1") + p/ ! 2 . 4. 
b+c+ + 
and c — p (A"' ~ A) — p 2 cot. g ( 5 '" + & - A'" A sin 1 ") + &c. &c. &c. . . . 
F 
= C = p(S — A) p s cot z Js-^-A(s-A)sinl"j + S JJ™J { — 
- A) 3 f (A" - A) 8 + (A'" - A) 8 + 
2 . 4 . 6 . 
&C.. 
adding up and taking a mean. 
C is therefore the quantity to be added to the mean z of the zenith distances, 
in order to have the zenith distance £ corresponding to the mean of the times. 
If the change of altitude were proportional to the change of time, C would be 
