MENSURATION. XXXI 



Area of ellipse ^=ir a b, 



= TV a- Vi^^^ 



=^ 77 a'^ cos (f>, if c = sin 4>- 



f. Surface of sphere, etc. 



r = radius of sphere, 

 ^j, ^2 = latitudes of parallels bounding a zone, 

 € = spherical excess of a spherical triangle 

 =^ sum of spherical angles less i8o°, 



Total surface = 4. ir r\ 



Surface of zone = 2 tt r^ (sin <^2 — sin 4>i), 



= 4. TT r- cos ^ (</)2 4" 4>i) sin J (</>» — <^i)- 



Surface of spherical triangle = r" e, for e in arc, 



= r"^ e/p", for e in seconds, 

 p" = 206 264.8", log p" = 5.3 1 442 5^3- 



g. Surface of right cylinder. 



r = radius of bases of cylinder, 

 /i = altitude of cylinder. 



Area cylindrical surface ■=z 2 it r 7i. 

 Total surface = 2 tt r (r -|- -^0- 



h. Surface of right cone. 



r = radius of base, 

 h = altitude, 

 s = slant height. 



Conical surface = tt r s = tt r (/i^ -^ r^)S 

 Total surface =: tt r (s -[- r). 



i. Surface of spheroid. 



a^ b =^ semi axes, 



e = eccentricity = {(a -{- b) (a — b)}ya. 



Surface of oblate spheroid = 2 tt a' 1 1 "h 3 e ^°^ \^^^/ > 



= 4^ a' {i - h e^ - -h ^' - -h ^' - ■ ■ ■)■ 



( arc sin e ") 

 Surface of prolate spheroid =z2'7?ab-{{i— e)--\- J- 



= 4-^ a b {1 - I <^ - ^ e' - ^\^, c' - . • ■)• 



* The logarithm in this formula refers to the natural or "Napierian" system. For areas of 

 zones and quadrilaterals of an oblate spheroid, see pp. 1-lii. 



