COBA 2 w A= cot p, or log cot - -)=/ cot p + 0; the logarithm being Naperian, and the angles being arcually measured. If t. be the length of the are of a degree of longitude at the equator, and if we now use degrees, and extend the preceding integration from to we have cot (45°- 1, r log cot — (4-0 L cot p; an equation from which p can be found for any two places, that is, the angle which the course in sailing from one place to the other makes with the meridian. And instead of r : L may be put its value 5729578. The distance from one place to the other on the rumb-line sailed over may be found from r cos p=r (A,-A,), which, when A, and A, are measured in degrees, becomes s cos a, neglecting the small correction for the earth's eccentricity.
The first of these processes can be done by 3fereator's chart, the principle of which, mathematically described, is as follows :-Let equal arca of longitude remain equal throughout the map, but as increments of latitude are to their corresponding increments of longitude as I to the cosines of the latitudes, let the differential triangle PQ n be similar in the chart to that on the sphere, which gives e d A : cos A, for the representation of d A on the chart, provided a represent the length of the degree of longitude on the chart. Ilenco a log cot (45° -4A) is the
length of A degrees of latitude measured from the equator ; and a table of values of log cot (45' - A) is called a table of meridional parts.
In such a chart all rumb-linea are projected into straight lines; but equal parts on any such straight line do not represent equal distances on the earth : and the distance sailed must be found by the formula in terms of the extreme latitude! and the angle of the course.