414 The Astronomer Royal on a Projection for Maps 



y shall =0 and -jL shall =0 (that is, so that the central parts 



of the map shall correspond exactly with the region about the 

 Centre of Reference), 



y = tan £ — 0— 2 cot^ . log cos x, 



and 



& 6 Q 



r—0 + y — tan~ + 2 cot^ . log sec <r, 



in which the logarithm is the Napierian or hyperbolic logarithm. 

 This equation entirely defines the nature of the Projection by 

 Balance of Errors. The numerical values of r, for a series of 

 values of 0, will shortly be given in a tabular form. 



10. In order to obtain a numerical estimate of the two errors 

 of a Projection, we must make use of the formulae, 



projected area _ r dr 

 original area "~ sin 0, dd ' 



projected breadth original length r X 



projected length original breadth ~~ sin0 dr * 



W 

 and for all the projections which we desire to compare, we must 



express r and -^ in terms of 0, and must substitute in these 



formulae. 



11. The Projections which I shall compare are the following 



(in the formulas, i/r is put for ■=) : — 



(1) The Projection with Equal Radial Degrees. In this, 



' o. a. sin 0' p. I. o. b. sin 



(2) The Projection with Unchanged Areas. In this, 



a • i P- a - -i V' b. 0. 1. 



r=2 sin ->/r, L = 1, *— j- x — — = sec 2 -Jr. 



o.u. p.L o.b. r 



(3) The Stereographic Projection. In this, 



r=2tanilr, £- =sec 4 -\lr, £—- x — - =1. 



o. a. T ' p.L o. b. 



(4) Sir H. James's Projection. Here 



_ 5 3infl p^ _ 25 (3 cos + 2 ) p.b . o.l. _ 3 + 2cos0 



3 + 2cos0' o.a~ (3 + 2cos6>) 3 ~' pl. X o7b.~ 3cos0 + 2' 



o 

 This projection fails when cos 0=—~, or = 131° 49'. 



o 



