From reading Leg 57 we know the following:
The first step is to compute the elapsed time. Subtract the previous date and time from the new date and time.
To compute a dead-reckoning position, it is a simple matter of vector
addition. On a plotting sheet -- ideally the one you used to plot
your last fix -- make a mark at the last known location of the boat.
From this mark, draw a line on vector 119.3. Multiply 10.3 knots by the time spent cruising -- 23.5 hours. On my slide rule, this comes to 242 nm.
Next, from the end of this vector, draw a second vector to represent the current.
Finally, determine the coordinates of the end of the final vector: lat = 53°38.4S lon = 74°44.6W
Note that as you sail near the poles, errors can accumulate rapidly. Bowditch addresses these points in some detail.
Next: Prepare a Plotting sheet