420 On a Projection for Maps by Balance of Errors. 



Thus, if we possess a map in which the meridians and parallels 

 are drawn through the circles and radial lines on one projection, 

 then for any other projection we have merely to draw radial lines, 

 and to describe circles with the radii given by the Table of 

 article 12, by the formula? of article 20, or by equivalent state- 

 ments, for that other projection ; and we can at once lay down 

 among the radii and circles of the second projection the inter- 

 sections corresponding to those of the first projection as Been 

 among its radii and circles. 



22. There is one projection in which the Meridians and Paral- 

 lels are described with comparative facility, because all are accu- 

 rately circular arcs, namely the Stereographic. This projection, 

 therefore, will be most proper for use as the standard projection, 

 by means of which any others may be drawn. As a termination 

 to this paper, I will here place the formula? required for drawing 

 a Stereographic Map with any Centre of Reference. 



Let a be the linear radius of the circle which would include a 

 hemisphere of the earth, /3 the radius of the proposed map, in 

 degrees. Let the Centre of Reference be in north latitude a. (If 

 in south latitude, it will only be necessary to invert the map.) 



(1) The linear radius of the entire map will be a . tan =•• 



S3 



(2) Through the centre of the map a line must be drawn as 

 polar axis. On this line will lie the centres of all the circles 

 representing parallels of latitude. 



(3) Let x be the north latitude of any parallel which is to be 

 drawn (a? being treated as an algebraically negative quantity for 

 parallels in south latitude). One intersection of the circle 

 representing this parallel, with the polar axis, will be north of 



the centre of the map by a . tan — ^— ; the other intersection 



will be north of the centre by a . cotan — ~. The centre of the 



circle will be north of the central point by half the sum of these 

 quantities, or by 



a x—u x + u 



•-: . cos « . sec — ^— . cosec— -c— • 



A 6 6 



The radius of the circle will be half the difference of these quan- 

 tities, or 



a x—u x + a 



jr . cos x . sec -■ . cosec — ^r- • 



The rules of algebraic signs are to be severely followed. 



(4) The north pole is north of the central point by 



