On the surface of a sphere a curve starts from the equator in a given direction and cuts all the meridians at the same angle. To ﬁnd its equation in latitude-and-longitude coordinates:
Let the loxodrome cross two consecutive meridians , in the points , ; let be a parallel of latitude; let , , , , all in radian measure; and let the angle ; then
hence , and , there being no integration-constant since vanishes with ; thus the required equation is
To ﬁnd the length of the arc : Integrate the equation
To illustrate numerically, suppose a ship sails northeast, from a point on the equator, until her diﬀerence of longitude is , ﬁnd her latitude and distance:
Here , and radians; radii. The latitude in degrees is .
If the ship set out from latitude , the formula must be modiﬁed as follows: Integrating the above diﬀerential equation between the limits and gives
hence , from which the ﬁnal latitude can be found when the initial latitude and the diﬀerence of longitude are given. The distance sailed is equal to radii, a radius being nautical miles.
Mercator’s Chart.—In this projection the meridians are parallel straight lines, and the loxodrome becomes the straight line , hence the relations between the coordinates of corresponding points on the plane and sphere are , . Thus the latitude is magniﬁed into , which is tabulated under the name of “meridional part for latitude ”; the values of and of being given in minutes. A chart constructed accurately from the tables can be used to furnish graphical solutions of problems like the one proposed above.