geodesic 

You can get pretty close by computing a geodetic straight line.  It might be possible to tweak that by using data from NGS's deflec program to simulate the affect of gravity to get a little closer.

To do a geodetic solution you can use any of a number of geodetic inverse/traverse programs compute the inverse to get a forward bearing at one end, then traverse out using a direct program using that forward bearing going 10km or the record distance.  Then obtain a new forward bearing by adding 180 to the back azimuth coming in and keep going.

A good example of a long line run is the C&GS running of the CA-NV oblique boundary.  They probably encountered some significant deflections of the vertical (>10 seconds) since they were running along and over some major mountain ranges. 

As I recall they triangulated to get the overall and then ranged out the line by double centering the whole line then correcting that for the small misclosing on the end.  Line was run from the north end near Lake Tahoe SE'ly to the south end at the Colorado River, and is one of the best examples I know of where someone really tried to lay out a true geodesic.

I would have to ponder the problem a bit to come up with a procedure to allow you to apply the geodetic deflections which we now have a model for but which were not available until the last 15 years or so.

If there are some modern positions on that CA-NV oblique boundary you might be able to treat it as a laboratory to test a computational method against and see how close you get.

- jerry l. wahl

PS You would also have to correct the record distances if they were based on ground survey by applying an elevation scale factor based on a weighted mean of the elevation profile along each segment of the line.