by Balance of Errors, 419 



portant, and the objections to the system of Equal Radial Degrees 

 are much diminished. And, as the Centre of Reference is so near 

 to the north pole, no serious discordance, probably no perceptible 

 discordance, will be produced, if we describe the Parallels as cir- 

 cles whose centres are in the north pole and whose radii increase 

 by equal degrees, and the meridians by straight radii from the 

 north pole with equal angles between each radius and the proxi- 

 mate radius ; and if we afterwards limit the map by a boundary- 

 circle whose centre is at longitude 1.6 h 20 m east of Greenwich 

 latitude 77° north. This construction would be extremely easy. 



20. Reverting now to the general theory, it appears that while 



n 



the Stereographic Projection, in which r = 2tan -, possesses the 



very great merit of being free from distortion in its small ele- 

 ments, yet a more acceptable map is given by advancing in some 



measure towards the Projection by Equal Radial Degrees, in which 

 a 



r=0=co xtan — . It is evident that this may be done conve- 

 niently by using larger numbers instead of the 2 and 2 which 

 occur in the stereographic formula. Thus we may conveniently 



B 3 B B 



use r=3taiiK, in which, Exaggeration = - — 5 . tan - . sec 2 ^, 



o sin o o o 



^ B B B 



Distortion = -. — -x . tan - . cos 2 ^. Or r = 4 tan -, which gives. 



sin 3 3 4 & > 



Exaggeration = ^^tan ^ . sec 2 ^ = sec^- . sec 4 -, 



Distortion = - — ^ . tan -. . cos 2 - = sec s . 

 sin# 4 4 2 



In either of these the Exaggeration is diminished, and Distortion 

 is introduced, but more in the second than in the first. 



21. I will now allude to the process by which any of these 

 Projections can be adapted to any Point of Reference whatever. 

 The process is in fact a transfer from one system of projection to 

 another system of projection, and is founded upon this theorem : 

 that if in one projection we describe a series of Circles whose 

 common centre is the Centre of the Map (corresponding to the 

 Point of Reference) having radii equal to values of r correspond- 

 ing on that projection to values of 6 which increase by uniform 

 quantities as 5° or 10°, and if we draw from that centre Radial 

 Lines at equal angles of azimuth ; and if we do the same thing for 

 another projection; then all the intersections of Meridians and 

 Parallels referred to the pole of the earth will occupy on one pi*o- 

 jection the same places, in reference to the circles and radial lines 

 above-mentioned, which they occupy on the other projection. 



