Qiiadrature of the Spherical Ellipse. S3 



sin ctt, cos CO successively, we shall obtain for the equations of 

 the spherical ellipse, 



sin^.r sin^ y . tan^^ tan^w , /^^ v 



-^-o- + -^-Q-i = 1» I — T- + ; — 5^ = 1- • (57.) 

 sin^a sin^/3 tanker tan^/3 ^ 



On the major and minor axes of the spherical ellipse as 

 diameters, let circles be described (see the fig. page 26), and 

 let a great circle be drawn through the centre of the ellipse, 

 making the angle (^ with the minor axis and meeting the cir- 

 cles in the points m and n\ through m and n let arcs of great 

 circles ?« s and n t be drawn at right angles to the spherical 

 axes OB, O A ; then as Om s is a right-angled spherical tri- 

 angle, sin ms =z sin « sin <p, in like manner sin tii = sin /3 cos <$>, 



s'm^jns sin^w^ , • p n i i i , 



or — r-3 1 . „ ^ =1 ; It follows then that 7ns and 7it are 



sm-^a sm-^/S 



the coordinates of a point on the spherical ellipse. 



Let p' be the central radius vector of this point, then sin^p' 

 r=sin^7« s + sin^n^ = sin^asin^ <p + sin^/3cos^ <^, but in (53.) 

 sin^ p = sin^ a sin^ <p + sin^ /3 cos^ <^, hence p=p'i or in (53.) p is 

 the central radius vector of a point of which the coordinates 

 are sin a sin <p, and sin |3 cos <^ respectively. 



XXIX. To find when y is a maximum. 



In this case -^ = 0, or from (49.) -^ -— = P -^. . (58.) 

 d\ ^ ^ dX dh dx^ 



Now p= ^/'^^o^K+¥m^i P= i/c^ + o^^os^A+^^im^; 



. ^P _ ~ {a^—l)^) sin X cos A 



hence ^ - —^^=^^^=^.^=, 



dP — (a^ — b'^) sin A cos A 



d A s/c"^ + a2 cos2 A + 6^ sin2 a 

 fl(2p _ _( g2-62) (fl^cos^A-^^sin^A) 

 ^ "" {a^cos^A + i^sin^A}* 



Making these substitutions in (58.) and putting tan « for — , 



tan /3 for — , we find 

 c 



tan a sec « sin « „ / en \ 



tan^ A = - — ^ ^ = -r—5 sec^ e, . . (59.) 



tan p sec /3 sin p 



a result analogous to (41.). 



XXX. To find a general expression for the value of w; as 



u^ _ TJ^p^ (a2 — 6^)^sinAcosA 



tan2u=p2= p2p- ^^2cos2A + 62sin2A)(c2 + a2cos2A+ h^sm^h)' 



I • o, a ^ 



we shall have, introducing the relations tan « =y, tan p = — , 



mil. Mag. S. 3. Vol. 25. No. 163. Jw^j/ 184.4. D 



