242 Mr. Ivory on the Method of folding the Length 



y the rectilineal distance of the two stations, or the chord be- 

 tween them ; to the difference of longitude ; a the equatorial 

 semidiameter, and e the excentricity of the elliptical meri- 

 dians. Further, let $ stand for the angle of depression of the 

 chord below the horizon of the first station : then y sin $ will 

 be the perpendicular drawn from the second station upon the 

 horizon of the first ; and the distance of this perpendicular 

 from the plane of the meridian of the first station, to which 

 plane it is parallel, will be equal to y cos <p sin m 9 because the 

 azimuth m is the angle which the projection of the chord upon 

 the horizon makes with the meridian. Now let x be the di- 

 stance of the second station from the polar axis of the sphe- 

 roid : then, co being the angle between the two meridians, it is 

 obvious that the distance of the second station from the plane 

 of the meridian of the first, will be equal to x sin co ; and in 

 consequence of what was before proved, we shall have this 

 equation, 



x sin co = y cos <p sin m. 

 Next let R be the radius of a sphere which passes through 

 both the stations, and touches the horizon of the first : then, 



sin * : =^r:' cc,s ^ = N/ ' 1 -ii?- 



If we conceive a plane to bisect the chord y at right angles, 

 this plane will cut off from the normal to the earth's surface at 

 the. first station, a part equal to the line R. Therefore R, like 

 the radii of curvature of the spheroid, will always be little dif- 

 ferent from a, the semidiameter of the equator ; and since y 

 is always a small part of R, or of a, we may substitute a for 

 R in the expression of cos <p, without danger of introducing a 

 sensible error in any case that can occur in practice. We 

 have likewise, from the nature of the elliptical spheroid, x = 



and thus we get, 



-e*sin*X' 



cos k' sin u y I yj. 



Vl-^sin 2 *' " « 4a"* 



sin m. 



Now, put sin — = Zr \ then, 



— V 1 — = 2 sin - '- cos ■£■ = sin 



a 4 a 1 2 2 



•: fi 



and the last formula will become, 



cos X r sin a> ♦ /> • 



- = sin p sin m. 



t/l— ei sin*>.' 



The like reasoning applied to the second station will furnish 

 another similar equation. 



Thus 



