Quadrature of the Spherical Ellipse. 



23 



XV. Let a cone lahose principal angles are supplemental he 

 cut by a concentric sphere^ the sum of the two spherical bases, 

 together "with twice the lateral surface comprised within the 

 sphere^ is equal to the surface of the sphere. 



XVI. We shall now proceed to establish some other ana- 

 logies between plane and spherical ellipses. 



To investigate the formula s =^fp dh + u for the rectifica- 

 tion of a plane curve, where p is the perpendicular from any 

 assumed point called the pole on a tangent to the curve, X the 

 angle between this perpendicular and any fixed line drawn 

 through the pole, u the portion of the tangent intercepted 

 between the point of contact and the foot of this perpendi- 

 cular. 



Fig. 3. 



Let Q be the centre of 

 curvature of the arc at A, 

 A B a tangent at A, O B a 

 perpendicular from the pole 

 O upon the tangent, O C a 

 perpendicular upon the ra- 

 dius of curvature Q A, then 

 A B = O C ; assume a point 

 a indefinitely near to A; let 

 abhe a tangent at a, O Z; a 

 perpendicular from O upon 

 this tangent, O T a perpen- 

 dicular from O upon the 

 radius of curvature Qa; 

 let O C cut the radius Qa 

 in t, and through i let t n be 

 drawn parallel to the radius Q A, then A a = differential of 

 the arc =ds = Aw + n a; now A n = Ct =:OT — O C=«6 

 — AB = du, and na=ntx Lain =p d\; hence 

 ds du 



d\~^ "^ dx 



(31.) 



It will be seen that in the proof of this theorem the radius 

 of curvature is assumed as being greater than p; should it, on 

 the contrary, be less, the expression becomes in that case 

 d s du 



dK~^ dK' 



hence generally 



=/><^ 



A + M. 



(31*.) his. 



It is obvious also that when the perpendicular p is equal to 



the radius of curvature, at that point of the curve — = 0, or 



d\ 



u is there a maximum or a minimum. 



