xliv GEODESY. 



« = —zf = (If) + (h/y + a/y + a/y + . . 



^ — 2 — /^ 2 ^4 ^ S ^ 16 ^^ 



^8 



I + y/i - ^■- 4 ^ 8 "^ 64 "^ 128 "^ 



The numerical values of the most useful of these quantities and their logarithms 

 are — 



log 

 rt; = 20 926 062 feet, 7.3206875, 



^ = 20855 121 feet, 7.3192127, 



e^= 0.00676866, 7.8305030 — 10, 



m = 0.00339583, 7.5309454 - 10, 



/2 := 0.00169792, 7.2299162 — 10. 



4. Equations to Generating Ellipse of Spheroid. 



With the origin at the centre of the ellipse, and with its axes as coordinate 

 axes, the equation in Cartesian co-ordinates is 



^ + -^ = 1, (i) 



a and 1/ being the major and minor axes respectively, and x and y being parallel 

 to those axes respectively. 



For many purposes it is useful to replace equation (i) by the two following : — 



X = a cos 6, 



y = 1/ shi 6, ^"^ 



which give (i) by the elimination of 6. This angle is called the reduced latitude. 

 See section 5, 



5. Latitudes used in Geodesy. 



Three different latitudes are used in geodesy, namely : (i) Astronomical or 

 geographical latitude ; (2) geocentric latitude ; (3) reduced latitude. The astro- 

 nomical latitude of a place is the angle between the normal (or plumb line) at that 

 place and the plane of the earth's equator ; or when the plumb line at the place 

 coincides with the normal to the generating ellipse, it is the angle between that 

 normal and the major axis of the ellipse. The geocentric latitude of a place is 

 the angle between the equator and a line drawn from the place to the earth's cen- 

 tre ; or it is the angle between the radius-vector of the place and the equator. 

 The reduced latitude is defined by equations (2) in section 4 above. The geo- 

 metrical relations of these different latitudes are shown in Fig. i by the notation 

 given below. 



