Can we still find a reasonable position?
Yes. We do a transferred position line plot. We shoot the sun as normal, work through the calculations till we have an intercept. Then from our deduced position we travel a certain distance to a new deduced position and shoot the same body again. Transferring the original position line the distance made good during the interval will give us a fair position.
As an example, in the South Atlantic I shoot the sun and from a deduced position of 40° 10' South and 10° 15' East I obtain a position line, after calculations, 10' away on a bearing of 50.06°. After steering 290° for 36 nautical miles a second sun sight is taken and the intercept is 8' away on a bearing of 10°. Transferring my first position line to my new deduced position I am able to get a position from two position circles.
The method has many opportunities for error. Course and distance made good can never be very accurate as wind and tidal drift can only be guessed at, the log is only approximate as to distance and the course steered will not be exactly correct either. The position so obtained will always be pretty "rough" - but some yachtsmen never use any other.