26 Mr. Ivory o)i deducing the Difference 



at either station is the horizontal angle contained between the 

 meridian and the vertical, or plane perpendicular to the earth's 

 surface, which passes through the other station. The azi- 

 muths m and m' on the surface of the spheroid become respec- 

 tively equal to the sphei-ical angles /a and ju-', when e-= 0, and 

 X = 0; but in every case, the sum of the first two may be 

 reckoned equal to the sum of the other two. Applying now 

 a well-known property of spherical triangles, we have this 

 formula, for finding the difference of longitude, viz. 



A — A' 



cos , , 



i an . — - = X cotan — - — , 



sin 



2 



which is a formula common to the sphere and to any spheroid 

 of small excentricity. 



Having cleared up this method of surveying, placed it on 

 its proper foundation, and made it general in its application, 

 we may now mention some other formulas derived from the 

 same principle. I shall putyfor the equal arcs ft— m and ?«'— jw.'; 

 and as the arcs are small, I shall suppose them equal to their 

 sines. Simplifying the value of y before given, we shall have 

 these equations, viz. 



f = e- cos A sin ?« x .' „, , 



■^ sin /J' ' 



' »i — m' ^ 



. rti-\-ni' , / ni — m A 



tan -^ = tan (^—^— +/). 



2 



Further, we have, f^mfi' _ 



e'x; 



cos A sin m 



and, by substituting this value in the equations (A), p. 433 of 

 the last Number of this Journal, and transforming them as in 

 the equations (B), we shall get. 



Tan u = r— -. , tan u' = 



sin >. tan m ' sin /.' tan vi ' 



cos (u+w) tan a' / / sin /3' \ 



cosM "~ tan A V cos A sin w sin A'/' 



cos (u' — ai) tan ^ / , f sin /3' 



tan X /- / sin /3' \ 



tan X' \ cos x' sin m! sin x/ * 



cos M 



We have now three different formulas for finding the longi- 

 tude, of which one is a deduction from the other two; and, as 

 they should all agree in giving the same result, they may serve 



as 



