Correction of Unequal Altitudes. 
89 
V . — Correction of Unequal Altitudes. 
It sometimes happens, in observing equal altitudes for the time, in rating a cliro- 
nometer, that the afternoon observation is lost by unavoidable circumstances or 
forgetfulness. In such a case, if the error of the watch be required, there is no help, 
but that of observing another or unequal altitude in the afternoon. To save the 
trouble of calculating each of the observations at full length, the following formula 
was investigated ; and as it may be useful to observers, we are happy to give it a place 
in our work. It is obviously applicable to other cases than the one above indicated. 
Let ZS be the corrected morning zenith distance, 
dz the difference of evening and morning, 
1* the horary distance from noon of morning observation, 
dp the difference of the horary distances from noon, 
TS the polar distance, 
PZ the co-lacitude; 
e . , , Sin 4 dz Sin (ZS 4. 4 dz) 
5 r Sin PZ Sin IS Sin (P-f-4 dp) 
The investigation is as follows. 
Cos ZS Cos PZ Cos PS 
1. Cos P = 
2. Cos 
Sin PZ Sin PS Sin PZ Sin PS 
_ . Cos (ZS + dz) Cos PZ Cos PS 
P 4- dp = — — ‘ 
Sin 
PZ Sin PS Sin PZ Sin PS 
„ „ Cos ZS— Cos (ZS + dz) 
3. Cos P— Cos (P + dp) — Sin PZ Sin PS 
_ Co s ZS— Cos ZS 4- 2 Sin ( dz Sin (Z-j- j dz ) 
Sin PZ Sin PS 
2 Sin 4 Sin (Z 4- 4 <**) 
Sin PZ Sin PS . 
But 4. Cos P-Cos (P + dp) = 2 Sin i dp Sin (P + 4 dp) 
2 Sin I dz Sin (Z 4- 4 dz) 
Therefore 5- 2 Sin 4 dp = yj,,' ~fZ Siu PS Sin (Pi- 4 dp) 
Z + i dz = evidently the mean of the zenith distances, and 
p + \dp= half the elapsed time. 
EXAMPLE. 
On the 27th May 1819, in latitude 31° 03',4 the following altitudes of the 
sun were observed at the time set opposite them. 
H< 
A.M. 58* 06', 9 at 21 3? 45,1 
P.M. 53. 57 ,4 „ 2. 28 50,7 
PS is found to be 68° 49', 6^ ^ ^ 
57,9 9,74719 
23,12 0,22678 
56,6 0,06720 
49,6 0,03035 
Sine 4 dz ■= 2° 
Sine (Z -f- 4 ^*) ^3- 
Ar. Co-sine (P 4 4 rf P) 
Ar. Co-sine PZ 58. 
Ar. Co-sine PS 68. 
Sine 2° 27'05" =8,63119 
Middle time, 0 
9 m 48.3s 
3 17,9 
23. 53 29 ,6 time of noon ; to be corrected by 
equation answering to change of declination. ... 
This formula also serves to find the change in altitude coiresponding to a given 
interval of time and vice-versa. 
