308 Col. Sir H. James and Capt. A. R. Clarke on Projections 



Putting r—6 = y, and giving to y only a variation subject to 

 the condition $y = when = 0, the equations of solution are 



Pp being the value of 2p sin 6 when 6=/3 ; hence 



sm^e^+sm6co S e^-y = e-s,me, . . (1) 



m = - 



(2) 



from which 



SB Of) 



y= — 0— 2cot-logcos^ + Ctan-+C'cot-, . (1) 

 Z Z Z Z 



0= cosec 2 f log cos f + \ C sec 2 f- i C cosec 2 ^.(2) 

 Z Z Z Z Z z 



Now y must vanish with 6 ; therefore C'=0, and, from (2), 



C= cot 2 § log sec 2 4 



which completely determines r. At the centre of the map, 

 where 0=0, 



dr _1 + C 



This quantity in the Astronomer Royal's paper is inadvertently 

 made =1, and consequently the computed Tables, pp. 415, 416, 

 417, are incorrect, and the Development appears under disad- 

 vantage. The limiting radius of the map is 



R = 2Ctanf- 

 Z 



This quantity does not increase indefinitely, but is a maximum 

 when /3=126° 24' 53" : for higher values of ft R diminishes. 

 When j3= 120°, R= 1-6007. When £=113° 30', which is the 

 limit of Sir H. James's map, R = 1*5760. 



Let us now compare this Development with the Equal Radial 

 and Sir Henry James's Projection. In order to do this, we must 

 suppose these to be drawn on such a scale that the limiting radius 

 shall be T5760, and then form the values of 



\dd V Vsin6> ) 



The values of this quantity, which call U, are given in the fol- 

 lowing Table : — 



