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Published online by Cambridge University Press: 03 November 2016
IT is well known that, while the shortest course between two points on a sphere is a great circle, the course which is easiest for navigation is a rhumb line, i.e. a course which cuts all the successive meridians at the same angle, so that the ship always steers in a direction making an angle of the same number of degrees with due north or due south. Such a course is represented by a straight line on a Mercator’s chart, which thus makes it appear more direct than it really is. On a great circle, though the course is the shortest, you are (except on the equator or a meridian) continually altering your direction as measured by the angle which it makes with the north. Between any two points there is thus a difference in favour of great-circle sailing; but to demonstrate rigorously the greatest value which this difference can possibly take is a somewhat difficult and attractive mathematical problem. I have heard it stated that it is not susceptible of rigorous mathematical treatment. The following is suggested as giving a complete solution of the question: it would be interesting to learn whether any shorter or more direct method is available, which makes no unwarranted assumption. For the abstract treatment of the problem (though indeed in the Pacific or the Southern Ocean fairly extended courses are practically available) we suppose the globe cleared of all such inconvenient encumbrances as continents or islands: moreover, we treat it as a perfect sphere; and to word the question more exactly we should say: Find two points such that the difference between the shortest great-circle course and the shortest rhumb-line course joining them is a maximum; for the great circle through two points will (in general) consist of a major and a minor arc, and on the other hand a rhumb line might start the longer way round the earth, or indeed make any number of revolutions about the pole before reaching its objective. In either case it is the shortest course of the kind with which we deal.
The substance of this paper was read to the Mathematical Section of the University of Durham Philosophical Society on December 4th, 1925.
page 138 notes * It might be thought that the great circle Joining two points on opposite meridians would be itself the limit of a rhumb line; but that is not so. You first go, say, due north, then due south. A rhumb line making a very small angle, Φ, with meridians will follow a meridian at first very closely, but ends by circling sharply round the pole.
page 138 notes † This will always be possible, with 
, for any course taken in one direction or the other.
page 140 notes * HK should terminate a little North of the 80° parallel. The dotted line should continue through the pole to K.
page 143 notes * The courses are plotted, in the diagram, on a stereographic projection of the hemisphere. I find that in this case the projections of the rhumb lines which cut the meridians at the same angle Φ will have points of Inflexion all lying on that diameter of the bounding circle which is inclined at an angle Φ, in the opposite sense, to the diameter through the poles. This property would make rather a pretty problem.