GEODESY. 



Values flf J(AX siti <f))\ 



Iv 



It appears from this table that the first terms of (2) will suffice in computing 

 the co-ordinates for projection of all maps on ordinary scales, and of less extent 

 in longitude than 2° from the middle meridian. For example, the value of x for 

 AA = 2°, and </> = 40°, and for a scale of two miles to one inch (i/i 26720), is 

 53.063 inches less i/iigoo part, or about 0.004 inch, which may properly be 

 regarded as a vanishing quantity in map construction. For the computation of 

 the co-ordinates given in the tables 19 to 24, where AX does not exceed 1°, it 

 is amply sufficient, therefore, to use 



X ■= pn AA cos </), 



y = \pr, (^^y sin 2<i>. 



(3) 



In these formulas and in (2), if AA is expressed in seconds, minutes, or degrees, 

 it must be divided by the number of seconds, minutes, or degrees in the radius. 

 The logarithms of the reciprocals of these numbers are given on p. xlvi. In the 

 construction of tables like 19 to 24, it is most convenient, when English units are 

 used, to express AA in minutes and x and y in inches. For this purpose, sup- 

 posing log p„ to be taken from Table 11, if s be the scale of the map, or scale 

 factor, equations (3) become — 



Co-ordinates x andj}' in inches for scale s. 



12 



X 



3437-747 



Pn S AA COS 4>, 



■2 



y = 7 \2 p„ s (AA)" sin 2(i>, 



(3437-747) ^" ^ ■> ^' 



AA in minutes ; 



log (12/3437.747) = 7.54291 — 10, 

 log (3/(3437-747)') = 3-4046 - 10. 



(4) 



Tables 19 to 24 give the values of x and j' for various scales and for the zone of 

 the earth's surface lying between o'^ and 80°. 



Nutnerical example. — Suppose ^ = 40° and A A = 15' j and let the scale of 

 the map be one mile to the inch, or .y = 1/63360. Then the calculation by (4) 

 runs thus : 



