Figure 9

Fig. 9.


Figure 10

Fig. 10.

The cosine-haversine formula serves the purposes of this problem very satisfactorily:

Hav z = hav (L~d) + cos L cos d hav h

which is derived from the well-known expression:

Cos z = sin L sin d + cos L cos d cos h

where z = zenith distance; L = the latitude; and h = the hour angle.

Solution

Dec. Arcturus 19° 42´ 29´´.
Lat. 55° 00 N. (Assumed). G. M. T. 20 d.  6 h. 20 m. 03 s.
R. A. M. ⊙︎  3 51 42
Acceleration  1 02
Lat. 55° 00´ 00´´. G. S. T. 10 12 47
Dec. 19  42  29 Long.  7 28 32
——————
L~d  35  17  31 L. S. T. 17 41 19
R. A. ⁜ 14 11 03
H. A. ⁜  3 30 16 W.
(Observer) 52 34 00

Lat. 55° 00´ 00´´ = cos. 9.75859
Dec. 19 42 29 = cos. 9.97378
H. A. ⁜ 52 34 00 = hav. 9.29244
9.02481 = nat. hav. .10588
nat. hav. 35° 17´ 31´´ .09189
z = 52° 48´ 35´´ = nat. hav. .19777
90 00 00
Computed altitude 37 11 25
Observed altitude 37 14 50
Altitude difference  =   3´ 25´´.


Figure 11

Fig. 11.

A ship’s position is usually obtained by plotting the lines of azimuth and the position lines much in the manner shown in the chartlet. The azimuth of the body at the moment of observation is readily taken by inspection from the azimuth tables or better still from Weir’s Azimuth Diagram, both published by the U. S. Hydrographic Office.

In order to get an intersection of two lines of position and thereby ascertain the latitude and longitude at once it is assumed that the observer took an observation of another star bearing S. 45° E., simultaneously with Arcturus.

When ordinary A.M. time sights are taken the resulting longitude establishes a north and south Sumner line but the latitude is by D. R.; at noon the latitude by meridian altitude establishes an east and west line but the longitude is by D. R. So it is with a Sumner line a position is established upon it but the position along it is by D. R. The latitude and longitude, however, can be obtained by a slight calculation without drawing the lines on the chart; that is, the most probable position. The altitude difference having been determined enter Table 2, Bowditch, using the azimuth, or its reciprocal as the case may be, as the course, and with the altitude difference as the distance, pick out the difference of latitude and the departure and apply them to the dead reckoning latitude and longitude as is the usual practice. The result is the most probable position (according to the D. R.) on the Sumner line.