Quadrature of the Spherical Ellipse. 31 



a formula for the rectification of curves on the surface of a 

 sphere analogous to (31.). 



As in none of the successive steps of the preceding demon- 

 stration is any reference made to the peculiar properties of 

 curves or cones of the second degree, it is clear that the pre- 

 ceding formula will hold for the rectification of any curve upon 

 the surface of a sphere, the intersection of this surface with a 

 concentric cone of any order, and as a curve traced liberd 

 manu on the surface of a sphere may be constituted the base 

 of a cone whose vertex shall be at the centre of the sphere, it 

 is plain that the above formula may be applied to the recti- 

 fication of any curve upon the surface of a sphere. 



Hence as an arc of any plane curve may be expressed by 

 means of a definite integral and a finite right line, so may the 

 arc of any curve described on the surface of a sphere be ex- 

 hibited by means of a definite integral and an arc of a circle. 



XXVI. To apply this formula to the rectification of the 

 spherical ellipse. 



Let a and b be the semiaxes of the elliptic base, r the cen- 

 tral radius vector drawn to the point of contact of the tangent w, 

 p the perpendicular from the centre on the tangent, u the in- 

 tercept of this tangent between the point of contact and the 

 foot of the perpendicular ; let a, j3, ^, -ccr, o be the angles sub- 

 tended at the centre of the sphere whose radius is c by the lines. 

 fl, bf r, py M, then 



a =i c tan a, ;6 = c tan /3, r=c tan p, p = c tan 'sr and «= P tan o. 

 Now in the plane ellipse p^ = a^cos^A. + h^ sin^A; hence 



tan^ 'sr = tan^ a cos^ A + tan^ j3 sin^ A, 

 and 1 = cos^A + sin^A. 



Adding these equations together, 



sec^ ■cT = sec^ a cos^ A + sec^ /3 sin^ A ; 

 dividing the former equation by the latter, 



, o tan^ a cos^ A + tan^ & sin^ A 



sec** a COS'' A + sec* p sm-* A ^ ' 



Substituting this value of sin ot in (47.), we obtain the equation 



/», /tan^ a cos^ A + tan^ /3 sin^ A . ^ , 



^=y^Wsec2acos^A + sec^/3sin^A + "- ' ' ^^^'^ 



XXVU. To investigate another forraula for rectification. 

 Assume sin^ p = sin^ a. sin^ <p + sin^ /3 cos^ ^ ; . . (53.) 

 hence cos^ f = cos^ a sin^ ^ + cos^ /3 cos^ <p. 



Substituting these values in (18.), we find 



=fd^\/ 



tan^ « cos^ ^ +. tan^ /3 sin^ ^ 

 sec^ « cos^ ^ + sec^ ^3 siu ^ * 



(54.) 



