Johann Bernoulli's work entitled
In superficie quacunque curva ducere
lineam inter duo puncta brevissimam ("To draw the shortest curve between
two points on any curved surface") appears as article number CLXVI in Volume
IV of the Opera Omnia.
A footnote at the beginning of the work states
that the information was communicated to Samuel Klingenstierna sometime
before the end of 1728, and that Klingenstierna later wrote the notes
on paper in order to preserve them. Thus the format resembles a student's
lecture notes, rather than a published article.
This site contains George Heine's very rough English translation of the work, together with illustrations from the original.
You are welcome to contact me with comments and suggestions.