Quadrature of the Spherical Ellipse. 



19 



and areas of those curves, nor of the striking analogies which 

 connect together the plane and spherical ellipse, an investiga- 

 tion of the same problem, conducted on different principles, 

 and leading to some very curious results, may not be unac- 

 ceptable to the mathematical reader. 



The method here pursued is founded on two general theo- 

 rems, which may be enunciated and proved as follows : — 



II. Theorem (1.). J%e ar^« A of any portion of a spherical 

 surface bounded by a curve may be determined by the formula 



A= / dai I cZo-. [sin<r],. , . . (1.) 



where o- is the arc of a great circle intercepted between a fixed 

 point P which may be termed the pole^ and any variable point 

 5 assumed within the curve on the surface of the sphere, p the 

 spherical radius of the curve measured from the pole and pass- 

 ing through the point s, w the angle, which the plane of the 

 great circle passing through the points P s makes with the 

 fixed plane of a great circle passing through the pole P. 



Fig. 1. 



Let O be the centre of the sphere, 

 P the pole, s the assumed point, P Q 

 the great circle passing through 

 them ; through P let a great circle 

 O P Q' be drawn indefinitely near 

 the former, d w being the angle be- 

 tween those planes ; through s let a 

 plane be drawn perpendicular to 

 OP, meeting the great circle O P Q' 

 in s'. Let a point u be assumed on 

 the circle P Q indefinitely near to s, 

 and through u let a plane be drawn perpendicular to O P, 

 meeting the great circle O P Q' in w' ; it is clear that the whole 

 area to be determined is the sum of the indefinitely small tra- 

 pezia, such as SM «'w', into which the required portion of the 

 spherical surface is thus divided. To compute the value of 

 this elementary area, we have s 5' = sin cr d<a, s u=id<r; hence 

 the area of the trapezium sus' u' = d 00 s'm <r d<j-; and the whole 

 area A round the pole P, and bounded by the curve, is there- 

 fore given by the formula 



/^2 IT /»|0 



A= / d 00 / c?cr[sin<r]. 



Integrating this expression between the limits p and 0, 



A=/ da)[l—cosp] (2.) 



III. Theorem (2.) To find an expression for the length of 

 a curve described on the surface of a sphere. 



C2 



