In the spherical triangle APV, P is the pole of the earth, V the pole of the vortex, A the point of the earth’s surface pierced by the radius vector of the moon, AQ is the corrected arc, and PV is the obliquity of the vortex. Now, as the axis of the vortex is parallel to the pole V, and the earth’s centre, and the line MA also passes through the earth’s centre, consequently AQV will all lie in the same great circle, and as PV is known, and PA is equal to the complement of the moon’s declination at the time, and the right, ascensions of A and V give the angle P, we have two sides and the included angle to find the rest, PQ being the complement of the latitude sought.
We will now give an example of the application of these principles.
Example.[10] Required the latitude of the central vortex at the time of its meridian passage in longitude 88° 50′ west, July 2d, 1853.
CENTRAL VORTEX ASCENDING.
| Greenwich time of passage | 2d. | 3h. | 1m. |
| Mean longitude of moon’s node | 78° | 29′ | |
| True"" | 79° | 32′ | |
| Mean inclination of lunar orbit | 5° | 9′ | |
| True"" | 5° | 13′ | |
| Obliquity of ecliptic | 23° | 27′ | 32″ |
| Mean inclination of vortex | 2° | 45′ | 0″ |
Then in the spherical triangle PEV,
| PE | is equal | 23° | 27′ | 32″ |
| EV | " | 7° | 58′ | 0″ |
| E | " | 100° | 28′ | 0″ |
| P | " | 18° | 5′ | 7″ |
| PV | " | 26° | 2′ | 32″ |
Calling P the polar angle and PV the obliquity of vortex.