Mr DAVIES on the Equations of Loci 



- 4. By means of (3) and (4) we can assign the polar equation the pro- 

 jection of any spherical curve upon a plane perpendicular to the axis, in 

 which the projecting point is situated. Let us take, for instance, the three 

 common projections, the orthographic, stereographic, and gnomonic, 



The Orthographic. Here b = infinite, and the equations are, 



v = asm<p, or ,~ ^ 



v ,V a * "* > ......... (5.) 



sin </> = - ; cos (p = r -_ , &c. I 

 a 1 



The Stereographic. Here b = a, and c = ; and 



sin 

 - 2-- 

 1 + cos (p 



sin f ,., 



v = - 2-- ; trom which 



r|2 



Insert this in cos 2 ^ +sin 2 0=1, and we get 



, 

 sm > = 



'_* 







The Gnomonic. Here 6 = 0, and c = a ; and the equation becomes 



a sin </) 



w = X = atana) 



cos^) 



.-. tand) = -; cosrf) = ===== ; and sind> = . = . . . (8.) 

 V^ + a 2 Vt> + a 2 



5. The different "globular projections" that have been proposed for the 



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construction of geographical maps, by taking as projecting points different 



* VThese two equations are foreign to the inquiry. The manner of their appearance 

 here is easily shewn; but the extent to which this paper runs, forbids my enlarging upon 

 this topic at present. 



