20 



Dr. Booth on the Rectification and 



Fig. 2. 



Let s and s' be two consecutive 

 points on the curve, Ps, Ps' two 

 great circles passing through these 

 points and the pole P, inclined to 

 each other by the angle d w ; through 

 s let a plane be drawn perpendicular 

 to O P, meeting the great circle P 5' 

 in u; then ultimately ss' u may be 

 considered as a right-angled triangle. 

 Hence (s ^Y = {s uf + C^' ufy but 

 s s' = dSyV s = f,^ su = s\r\ p d Wi s' u 

 ^dpi or {dsf={dpY +{smpdcoy. 



Integrating this expression and taking the limits p^, pQy 



IV. Def. A spherical ellipse is the curve of intersection of 

 a cone of the second degree with a concentric sphere. 



Let 2 ex. and 2 /3 be the greatest and least vertical angles of 

 the cone, which may be termed the principal angles of the 

 cone, the origin of coordinates being placed at the common 

 centre of the cone and sphere, and the real axe of the cone 

 assumed as the axis of z meeting the surface of the sphere in the 

 point P, the centre of the spherical ellipse, which point may be 

 taken as the pole. Let the mean axe of the cone be in the 

 plane of ,2? 2?, the least in that of^ z ; p being the arc of a great 

 circle drawn from P to any point Q of the ellipse, w the angle 

 which the plane of this circle makes with the plane of x z, in 

 which the semiangle « of the cone is placed, then the polar 

 equation of the spherical ellipse is 



c2 ,,\ cin* fi\ 1 



COS^ CO 



sm^o) 



tan'^ « tan^|3 tan^p* * ' • • v v 



To show this, through the point P let a tangent plane be 

 drawn to the sphere, intersecting the cone in an ellipse, which 

 for perspicuity may be termed the elliptic base of the cone ; let 

 the great circle passing through P and Q cut this ellipse in 

 the central radius vector r, a and b being the semiaxes of this 

 section, and c the radius of the sphere ; then we have from the 

 common equation of the ellipse, 



cos^co 



+ 



sm" (o 



1 



a^ b^ 7 



00 being the angle between r and a, but a 



r — c tan p ; making these substitutions, 



cos'^ CO sin^ CO 1 



.2» 



c tan oiib = c tan /3, 



+ 



tan^ a tan^ /3 tan^ p 



