382 Mr. Rumker’s new Method of 
sions of both bodies, and computed thence, with true dif- 
ference of right ascensions and true declinations, the true di- 
stance, and withthe apparent difference of right ascensions 
and apparent declinations, the apparent distance, and the 
differences A‘— A of these apparent and true distances for 
three successive equally distant periods ¢,, ¢,, ¢3, and denoted 
by «5 w2, wa, the remainders left by a subtraction of these 
differences A‘— A from one another. 
Suppose now w, = A + Bt, + Cz#,’, a, = A+Bé, +Ct,’, 
and a, = A+Bz#,+Cz,?, then is, © 
w, (t3—ty) + &(t;—ts) + 3 (ta—t)) 
(t.—7,) (t3—t,) (tg—ta) 
Parra 
t,—t, 
C= 
p= C(¢,+7¢,) and A = w, —¢,(B+#, CQ), 
whence any other # = A + Bt+C# for any given ¢ may 
be found, provided A is assumed proportional to the given 2; 
Then w+ sum of apparent semidiameters applied to the distance 
taken from the Naut. Almanac for the time reduced for longi- 
tude to the first meridian, gives the apparent distance of limbs, 
which by a comparison with the observed distance will show 
the error of the assumed longitude. This rather laborious 
proceeding may be simplified by taking also from the Nauti- 
cal Almanac for the same reduced time, together with the 
other elements, the true distance of the centres, and finding by 
differ. R cosin declin. 
sine distance 
the sun, as well as for the moon; then a subtraction of the 
parallactic angles from the angles of position will give the 
above-mentioned angles S and M, whence will be found, 
A’—A = (a'—¢') cos M—(g— =) cos S+ third correction, 
where S and M are considered acute; any doubts whether the 
angles of position are obtuse or acute, are easily decided, and 
the reduction of the refraction to the points of contact is ac- 
complished in the same manner as before. This method offers 
advantages when by a series of observations the longitude of 
a place on shore is to be determined. At sea it would be 
unwise to neglect the opportunity of observing the altitudes 
above the visible horizon, considering that the latitude enter- 
ing into the calculation of the altitudes rests upon no firmer 
base than the contemporaneously observed altitudes, and re- 
quires moreover a very unsafe reduction by dead-reckoning 
to the time and place of lunar observation, not to mention 
, the sines of the angles of position for 
