A Geodesic on a Spheroid. 107 



Write in this sin //=sin y V i— cos- //y sin- «=/) sin i-, 

 then cos- it — cos- //,( sin- a = i — si n- // — ( i — />-) 



=/- cos- »' 

 . • . ils-=a- dv- (I — ('- + ^'" /- sin- c) 



.s :=\ a^/ i—c- + c^ p^ sin^ y dv 



.-. s = a Vi-g--ht-y Jy^i- ^_J^"^,2^o cos2 „ j„. 



Draw then an clhpse, semi-major axis a ^ i—e^ + c-J>-, that is 



ep 



a^/ 1—6'- cos- //q sin- a, eccentricity , that is 



• ^i—c- + c-p- 



yi — cos- ;/o sin- a 

 1—6'- cos- II (j sin- a 



then arc of geodesic from point, eccentric angle //u to point, 



eccentric angle //, where tan (colatitnde) = - cot (eccentric angle), 



is arc of this ellipse from point, eccentric angle r^ to point, eccentric 



angle c, 



where sin // = sin r^i — cos- //,j sin- a- 



Note the semi-minor axis of this ellipse 



r. . — / ^'' (I- 



= <?v/i — t- COS- //osm- a X ^ I — y _ .3 



— COS"* ;/o sm- a) 



cos- //(, sm- o 

 = a Vi— t'"^ 



= (', semi-minor axis of spheroid. 



4. The relation between (p and // is 



d<f- COS- // i*—^ cos- //— I ) = (l— c'-COS- //) dll-. 



Take the above ellipse and let .v be the angle made with the major 

 axis by the perpendicnlar on the tangent at the point whose 

 •eccentric angle is r, 



then tan i- =— tan .v, where a^ is semi-major axis of ellipse. 

 "1 

 Turn the relation between and // into one between <!> and .v, and, 

 after some reduction, we tind 



1, '^ - a- (i-p-) {i-c^) dx^ 



a{' {i — k- sin^ .v) (i ^ siii^ -i')^ 



, , . i^ • -.^ ,- 11- • / I — cos- //a sm- a „ , 



vhere k is eccentricity ot ellipse, i.e., e / . 5-^^ — r-^— ,ancl 



V I— 6'- cos- //o sm^a 



Ov^^os 



:<^ 0° 



Q^, :J^ 



