Quadrature of the Spherical Ellipse. S7 



therefore s' — s — u » , (35.) 



Hence we may take on an elliptic quadrant two arcs mea- 

 sured from the extremities of the minor and major aXes respect- 

 ively, whose difference shall be equal to a right line. 



XVIII. It is not difficult to show that the extremities of 

 these arcs are the points of intersection of the given ellipse 

 with two hyperbolas having the sa.me foci as the given ellipse, 

 one passing through the extremity of the arc measured from 

 the minor axe, whose axes A, B, are given by the equations 



A = fl!^sinA, B^aecosX;. . . . (36.) 

 the other passing through the extremity of the arc measured 

 from the major axe, its semiaxes A', B' being deduced from 

 the equations 



V^ ggcosX B»^ bee.m\ 



Vl-e^sin^x* ^/l-e^sin^A** " ^ -* 



XIX. To determine the general value of m, 



dp d. . - a e^ sin A cos X 



We may hence deduce some remarkable relations between w, 

 a, b, A, B, A', B' ; for by the help of the preceding equations 

 it is easily shown that 



r> Til r 



au=AA', 6«=BB', t~I7 = —• * • • (38-) 



Let 2 9 and 2 d' be the angles between the asymptots of those 

 hyperbolas, then 



Ian fi = -J- = cot A, and tan 6' = — tan A; . (39.) 



hence tan d tan & = — , 



a 



a result independent of X. 



XX. Let r' and r" be the semidiameters of the ellipse mea- 

 sured along these asymptots, then 



cos^fl sin^e _ J_ 

 «2 + 62 ~ rf^J 



or putting for cos 6, sin S their values deduced from (39.), we find 



■2 __ ^^ b'^ _ f^ b^ 



"~ a^ cos^ X + 6^ sin^ X ~ p^ 

 In like manner it may be shown that 



r"2 = c^ cos^ X + 6^ sin^ X = js^ ; 



hence r^ r" = ab, (40») 



a result also independent of X. 



We have thus the remarkable result that the segments of 



