Quadrature of the Spherical Ellipse. 85 



Substituting the values of sin A, cos A, thence derived in (61.) 

 and (62.), there results 



tan^ A = tan a sec a (sin « — sin /3) 

 tan^ B = tan /3 sec /3 (sin a — sin /3) . ^ 

 tan^ A' = tan a sec a (sin « — sin /3) ^ ' ' * ^ 

 tan^ B' = tan (3 sec /3 (sin a — sin |3). 

 hence A = A', B = B' ; or when o is a maximum the two hy- 

 perbolas coalesce, and the arcs of the ellipse have a common 

 extremity, or constitute the quadrant, and this point may be 

 termed the point o^ circular section. 



XXXII. To find the value of y when y is a maximum ; as 

 tan w tan a cos /3 = tan A tan A' = tan'^ A = tan « sec a (sin « — sin j8) 



tan u = sec « sec (8 (sin a — sin /3) {'^^') 



XXXIII. To find the values of the arcs of the asymptotic 

 circles to the hyperbolas contained within the spherical ellipse. 



The asymptotic circles to the spherical hyperbola are the 

 great circles whose planes are parallel to the circular sections 

 of the cone, of which these hyperbolas are sections. 



Let 2 5 be the angle between the great circles which con- 

 stitute the asymptots of the hyperbola passing through the ex- 

 tremity of the arc measured from the minor axe ; then, as the 

 asymptotic circles are parallel to the circular sections of the 

 cone, and whose principal semiangles are a' and /3', of which 

 the given hyperbola is a section, we shall have (see (21.)), 



sur 5 = . J , ; 



but it may be shown that 



sin /3' = cos \ and sin a' = cos A ; 



hence sln^Q = — h-t- 



cos-* A 



Substituting for cos^ A its value derived from (61.), we find 



A cos 3 , 



tan 9 = cot X {QQ.) 



cos a. ^ ' 



Eliminating 9 between this equation and the equation of the 



... cos^S sin^fi 1 , . 



ellipse -T-^5 [- -.on — -^-o-j there results 



' sui^a sin'' p sin'^p 



tan^ a cos^ A + tan^ /3 sin^ A _ sin'^ a sin^ jS ^ . ' 



sec^ a cos^ A + sec''^ /3 sin^ A sin^ p ' 



but it has been shown (51.) that the first member of this equa- 

 tion = sin^-BT. 



Making this substitution, 



sin « sin /3 ,^^ . 



sin = -. (68.) 



' sin«r ^ ' 



D2 



