1 



GEODESY. 



The values of A/' for intervals of lo", 20" . . . 60", and for 10', 20' 

 are given in Table 18 for each degree of latitude from 0° to 90°. 



8. Radius- Vector of Earth's Spheroid. 



p = radius-vector 

 = a (i — 26- sin- ^ -\- e'^ sin'*' (^)* (i — e^ sin^ <^)~*' 



60' 



log p = log 



(7 (2 — e^ 

 ' + S/i - ' 



= -|- /i, (;« — ti) cos 29!) 



— ^ /It (w- — ;/-) cos 4<^ 

 -]- ^ /A (w* — «") cos 6<^ 



For the adopted spheroid 



log (p in feet) = 7.3199520 -|- [3.86769] cos 2^ 



— [1-2737] cos 40, 



the logarithms for the terms in ^ corresponding to units of the seventh decimal 

 place. Thus, for cf) = o, 



logP = 7-3199520 



+ 7373-8 



— 18.8 



= 7.3206S75 = log a. 



9. Areas of Zones and Quadrilaterals of the Earth's 



Surface. 



An expression for the area of a zone of the earth's surface or of a quadrilateral 

 bounded by meridians and parallels maj'^ be found in the following manner : — 



The area of an elementary zone dZ, whose middle latitude is cf> and whose 

 width is p„ d4>, is (see Fig. i), 



dZ = 2 TT r p,„ d(ji 



= 2 TT p,„ p„ COS d^. 



By means of the relations in section 6 this becomes 



,7 2/ 9A cos cjy d^ 



^ ^ (i — c'' sin- (fjy 



2 IT a 



I — e^ d (e sin (/>) 

 e (i — e'^ s\\\^ cji)" 



(0 



The integral of this between limits corresponding to <^, and cf),,, or the area of a 

 zone bounded by parallels whose latitudes are ^i and ^2 resjDcctively, is 



Z = TT a' 



1 — e^ 



e sin <^2 



e sin <^, 



I — e^ svvi' (p., I — ^"■^ sin^ (jf). 



1 (i + ^ si" <^2) (i — ^ s in <;^i) 



+ h Nap. log } — — -. — r\~7 — \ • — r\' 



. ' ^ '■ ^ (i — e sin </).j) (i + ^ sin 4>]) J 



y (2) 



