120 TJ. 8. COAST AND GEODETIC SURVEY. 



is taken as unity. The half zone with a base limited by the 

 parallel of latitude <p has the area 7r(l — sin (p).. It is pro- 

 jected upon the portion of the plane PTJDTJ^ which the 

 chord 11 V divides into two segments of circles; the one 

 UPTJ* is the difference between the sector OUPTJ^ meas- 

 ured by 2 OP^ times the arc UPU' or by tt — 2«p', and 



the triangle OUU', which is measured by 2 OTJxOWx 



sin Z VOV or by sin 2(p' ; the other segment is the difference 

 between the sector STJDU' and the triangle SUU'; the 

 angle USU^ is equal to 2<p\ and the radius SU of the par- 

 allel is equal to ^2 cot <p', so that the area of the segment 

 is equal to (2<p' — sin 2^') cot^ ^'. B^r equating the area 

 of the zone to the area of the projection of the same, we 

 obtain the relation 



TT — irsin^ = 7r — 2^' — sin 2^' + (2<p^ — sin 2^') cot^ <p' 

 or 



IT . SlD.2(p^ — 2(p' C03 2<p' 



2 ^"^ ^ 1-C0S2*,'- 



According to the second condition, the area of the segment 

 OPOP* ought to be equal to that of the lune formed by the 

 central meridian with the meridian of longitude X. The 

 angle PT^' is the angle X', so that TP=^2 cosec X'. The 

 area of the segment OPGP' is equal to the area of the sector 

 TPOP'f minus the area of the triangle TPP', 



TPOP' ^ItP'x arc POP' 



=^X2cosec^X''X2X' 

 = 2X'cosecn' 

 ATPP' = ItPx TP' sin ZPTP' 



= 2X2 cosec^ X' sin 2X' 



TPP' = cosec^X' sin 2X'. 

 Hence for the area of the segment we obtain 



0PGP' = 2\' cosec^X'-cosec^X' sin 2X'. 



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