by Balance of Errors. 4*1 1 



pressing the "radii of map-circles " in article 2 in terms of the 

 ■ great-circle radii on the earth," which will make the total evil, 

 represented as has just been stated, the smallest possible. 



5. Now let I and b be the length and breadth of a small rect- 

 angle on the earth's surface, and suppose that the length and 

 breadth of the corresponding rectangle on the map are /+S/and 

 b -f- 8b, and neglect powers of 81 and 8b above the first. (Although 

 this docs not apply with algebraic correctness to very great 

 change of area and distortion, yet it will be found by the result 

 that the theoretical failure introduces no practical inconve- 

 vience.) Then the Change of Area 



_ projected area _ _ (1+81) .{b + 8b) 81 8& 



~ original area ~~ lb ~ I b 



And the Distortion 



_ projected length original breadth __-. _ 1+81 £ _ 

 ~" projected breadth original length ~~ b + 8b 1 



_8l_8_b 



" I b' 



The sum of their squares, or I "T + T" ) + \T ~~ X j > 1S 



And therefore we may use (j) -f \-r ) as the measure of the 



evil for each small rectangle. 



6. Let be the length, expressed in terms of radius, of the 

 arc of great circle on the earth connecting the centre of the small 

 rectangle with the Centre of Reference; r the corresponding 

 distance on the map, expressed in terms of the same radius, of 

 the projection of the centre of the small rectangle from the 

 centre of the map ; the object of the whole investigation is to 

 express r in terms of 6. Let the length of a small rectangle on 

 the earth be 86, the corresponding length on the map 8r. Also 

 let <f> be the minute angle of azimuth under which, in both cases, 

 the breadth of the rectangle is seen from the Centre of Reference 

 or the centre of the map. Then we have 



1=80, l + 8t=8r, 8l=8r-80; 



b = <f>.mi0, b + 8b = <f>.r, 8b = <f>. {r- sin 0); 



(fMIHi-OM^- 1 )- 



This quantity expresses the evil on each small rectangle. The 

 2E2 



