for Maps applying to a large extent of the Earth's Surface. 309 



0. 



Balance of Errors. 



Sir H. James's. 



Equal 



Radial. 



6. 



U. 



U sin e. 



U. 



U sin0. 



U. 



Usinfi. 



• 

 



•1168 



•0000 



•1182 



•0000 



•0836 



•0000 



o 







5 



•1164 



•0101 



•1172 



•0102 



•0832 



•0072 



5 



10 



•1149 



•0200 



•1147 



•0199 



•0819 



•0142 



10 



15 



•1126 



•0291 



•1106 



•0286 



•0799 



•0207 



15 



20 



•1093 



•0374 



•1050 



•0359 



•0771 



•0264 



20 



25 



•1052 



•0444 



•0980 



•0414 



•0737 



•0311 



25 



30 



•1002 



•0501 



•0897 



•0448 



•0696 



•0348 



30 



35 



•0945 



•0542 



•0804 



•0461 



•0651 



•0373 



35 



40 



•0881 



•0566 



•0704 



•0453 



•0603 



•0387 



40 



45 



•0812 



•0574 



•0601 



•0425 



•0553 



•0391 



45 



50 



•0740 



•0567 



•0499 



•0382 



•0506 



•0387 



50 



55 



•0667 



•0547 



•0404 



•0331 



•0464 



•0380 



55 



60 



•0598 



•0518 



•0325 



•0281 



•0432 



•0374 



60 



65 



•0536 



•0486 



•0270 



•0244 



•0418 



•0379 



65 



70 



•0491 



•0461 



•0251 



•0236 



•0430 



•0404 



70 



75 



•0471 



•0455 



•0285 



•0275 



•0479 



•0463 



75 



80 



•0492 



•0485 



•0391 



•0385 



•0582 



•0573 



80 



85 



•0575 



•0573 



•0597 



•0594 



•0759 



■0756 



85 



90 



•0752 



•0752 



•0937 



•0937 



•1041 



•1041 



90 



95 



•1068 



•1064 



•1461 



•1456 



•1469 



•1463 



95 



100 



•1593 



•1569 



•2240 



•2206 



•2099 



•2067 



100 



105 



•2412 



•2359 



•3371 



•3256 



•3013 



•2910 



105 



110 



•3784 



•3552 



•5002 



•4700 



•4329 



•4068 



110 



115 



•5904 



•5352 



•7350 



•6661 



•6223 



•5640 



114 



From these quantities we may infer the values of 

 ^Usin<9.^ = M 



I 



for the different systems ; they are 



Sir H. James's . . . M = 0*1718, or as 109*5 

 Equal Radial .... = 0-1643, „ 1047 

 Balance of Errors . . =0*1569, „ 100*0 



By inspection of the Table it will be seen that from 0° to 45° 

 the Equal-Ptadial Projection has the advantage; from 50° to 80° 

 Sir H. James's Projection has the advantage; from 80° upwards 

 the Balance of Errors has the advantage. 



The first of these projections may, however, be greatly im- 

 proved. The general expression for the radius, namely 



&sin 

 T ~ A+cos#' 



involves two arbitrary quantities, and we may so dispose of them 

 as to render the integral M an absolute minimum ; that is, 



J LU + COS0 / \ (/*+cos0) 2 / J 



