Article 35
The Loxodrome.

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 AM, AN in the points P, Q; let PR be a parallel of latitude; let OM = x, MP = y, MN = dx, RQ = dy, all in radian measure; and let the angle MOP = RPQ = α; then

tanα = RQ PR , but PR = MNcosMP,17

hence dxtanα = dysecy, and xtanα = gd1y, there being no integration-constant since y vanishes with x; thus the required equation is

y = gd(xtanα).

To find the length of the arc OP: Integrate the equation

ds = dycscα,  whence s = ycscα.

To illustrate numerically, suppose a ship sails northeast, from a point on the equator, until her difference of longitude is 45, find her latitude and distance:

Here tanα = 1, and y = gdx = gd 1 4π = gd(.7854) = .7152 radians; s = y2 = 1.0114 radii. The latitude in degrees is 40.980.

If the ship set out from latitude y1, the formula must be modified as follows: Integrating the above differential equation between the limits (x1,y1) and (x2,y2) gives

(x2 x1)tanα = gd1y 2 gd1y 1;

hence gd1y2 = gd1y1 + (x2 x1)tanα, 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 (y2 y1)cscα radii, a radius being 60 ×180 π nautical miles.

Mercator’s Chart.—In this projection the meridians are parallel straight lines, and the loxodrome becomes the straight line y = xtanα, hence the relations between the coordinates of corresponding points on the plane and sphere are x = x, y = gd1y. Thus the latitude y is magnified into gd1y, which is tabulated under the name of “meridional part for latitude y”; the values of y and of y 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.

Prob. 103.
Find the distance on a rhumb line between the points (30 N, 20 E) and (30 S, 40 E).