Qtiadrature of the Spherical Ellipse. 37 



, tan^? tan^M . , 



and - — 5 — h - — 5— = 1 m the ellipse ; 



taira tair/3 ' 



, tan H tan m 

 hence = ii 



tan a tan /3 



and by the rules for right-angled triangles, 

 tan^H . 2 , 

 tan^ a ' 



. 9 - , tan^ >j tan^ R sin^ X 



Ql» Sin <a ^Z — ^^ - . 



tan^ /3 tan^ a cos^ a + tan^ /3 sin^ a* 

 Eliminating ^' between this equation and the polar equation 

 of the spherical ellipse, 



tan2r'= tan^acos^X + tan2/3sin^A; . . . (73.) 



as c = -^ — A, 



C2 " _ tan^ a tan^ /3 



~ tan^ a cos^ A + tan^ /3 sin^ a' 



or tan r tan r' = tan a tan /3 (V*-) 



XXXVI. Resuming the values of the angles which the 

 asymptots of the spherical hyperbolas, as also the diameters 

 of the ellipse through the points m and /a on the circle make 

 with the major axe, we find 



cos /3 



tan 3 = ~ cot A, tan d = cot A 



cos a 



,, tan/3, ., tan/3 



tan 6' = tan A, tan d' = tan A 



tan a, tan a 



(75.) 



We may here perceive a remarkable interruption of the 

 analogy which has been found hitherto to exist between the 

 properties of plane and spherical conies, while in the plane 

 section the asymptots to the confocal hyperbolas coincide with 

 the diameters drawn through the points m if., as is also true of 

 the spherical hyperbola adjacent to the major axe ; the asymp- 

 tot of the hyperbola nearer the minor axe does not coincide 

 with the diameter through the point m', in other words, while 

 6' = ^', fl is not equal to ■&. 



XXXVII. Two tangents being drawn to the spherical 

 ellipse at the point of circular sectioti and produced to meet 

 the adjacent axes, to find the values of those circular arcs. 



Let 13 be the coordinate of this point along the axis of Y, 

 and z the point in which the tangent arc t cuts the minor axis ; 

 let oz = ^, then tan >] tan ^ = tan2/3; and as t, x and (?—>;) 

 are the sides of a right-angled spherical triangle, 

 cos T = cos x cos (^ — »)) = cos ^ cos 5 cos »j + cos a; sin ^ sin >;. 



