S* Dr. Booth on the Rectification and 



and those given in (11.) and (12.), 



e^ sin a sin X cos X . __ v 



tan = — ■ o • o • • • (""•) 



V 1— e^sin^A V 1 — sin^esin^X 



XXXI. Having ah'eady exceeded the conventional limits 

 of a mathematical paper in this Journal, it may suffice to give 

 the enunciations of a few theorems on the spherical ellipse 

 analogous to those which have been already established on 

 the plane ellipse, postponing their discussion to a future oc- 

 casion. 



Through the extremities of the arcs of the spherical ellipse 

 two spherical hyperbolas may be drawn having the same focus 

 as the spherical ellipse; calling the axes of the one nearer to 

 the minor axe 2 A and 2 B, the axes of the other passing 

 through the extremity of the arc measured from the major axe 

 2 A' and 2 B', we may with little difficulty establish the fol- 

 lowing relations: — 



o. sin^esin^A „„ sin^scos^X ,^, , 



'"" ^ = l-si„^.sin^V '"" B = i_sinnsin^x - ' («'•' 



,a„>> A' = ."■"'^'^"f^ tan^ B' = ^'''"^'"'"'^f \ (62.) 

 l—e^ sm^ A, I — e^ sm"^ A ^ ^ 



We may hence show that 



tan u tan j8 cos « = tan B tan B'"\ 

 tan u tan a cos /3 = tan A tan A' J ' ' ' * \ 'I 

 tan B tan B' _ tan /3 sec /3 

 tan A tan A' ~ tan a sec a' 

 results analogous to (38.). 



In the spherical hyperbola e' being the eccentricity, we shall 

 find 



^ 2 , tan''' A + tan^ B 



^""^= 1-tan^B ' 



A and B being the semiaxes, while s being the eccentricity of 



the spherical ellipse whose semiaxes are « and /3, 



g tan^a — tan^/S 



tan^ 6 = — :; c, » ' 



1 + tan^/S 



Let g" be the eccentricity of the hyperbola whose semiaxes are 

 A' and B', we shall find, putting for A, B, A', B', their values 

 given above, 



s = s' = e". 



When y is a maximum we have found for the correspond- 

 ing value of tan^A the expression 



sin a o 

 - — ^zsec^e. 

 sm/3 



