30 Mr. Ivory on the shortest Distance 



the masts and ship appeared as if they were separated, then 

 the masts touched each other, and afterwards rapidly increased 

 in length, presenting quite a distorted appearance. The sea 

 looked as if it were on a level with the eye; and although it was 

 four or five miles off, the ripple upon the water was distinctly 

 visible. This refraction was probably caused by the difference 

 of temperature between the lower and upper stratum of air 

 (18° of Fahrenheit), which produced a medium of varied 

 density; and from the extreme evaporation over the sea, the 

 refractive power there was small, but gradually increased to 

 the point of observation, which possibly was its utmost limit, 

 as after we ascended a few yards this unusual refraction or 

 mirage went off, and the vessels assumed their usual ap- 

 pearance. 



[To be continued.] 



VI. A direct Method of finding the shortest Distance between 

 two Points on the Earth's Surface when their Geographical 

 Position is given. By James Ivory, Esq. M.A. F.R.S. fyc* 



r T , HE geometers who, supposing that the earth is an oblate 

 * elliptical spheroid of revolution, have investigated rules 

 for computing the shortest distance between two points on its 

 surface, usually assume that there are given the two latitudes 

 and the angle which the geodetical line makes with the meri- 

 dian of one of the points. The direct problem for deducing 

 the geodetical distance from the geographical position of the 

 two points has not hitherto been solved. In the Conn, des 

 Terns for 1832, M. Puissant has given formulas which supply 

 this defect. But his method merely consists in deriving from 

 the two latitudes and the difference of longitude, what is 

 necessary for applying the formula for the shortest distance 

 published long ago by M. Legendre. I shall here shortly 

 explain a different solution of the problem, which I lately ob- 

 tained; according to which the shortest distance of two points 

 on the earth's surface is expressed by means of their latitudes 

 and the inclination to the equator of the great circle of the 

 celestial sphere that passes through them. 



The radius of the equator of an oblate elliptical spheroid 

 of revolution being represented by unit, and the semipolar 

 axis by V 1 — e\ let X denote the latitude of a point on the 

 surface, and i> the longitude reckoned from a fixed meridian : 

 then, if x, y, z be the three coordinates of the point, the ori- 



* Communicated by the Author. 



gin 



