Royal Society. 5$ 



cable to, — 1st, the quadrature, and 2nd, the rectification of the sphe- 

 rical ellipse. 



1st. The quadrature of the spherical ellipse is reduced to the 

 calculation of a complete elliptic function of the third order, whose 

 parameter and modulus are quantities essentially related to the 

 cone; its parameter being the square of the eccentricity of the 

 ellipse, whose plane is at right angles to the axis of the cone, and 

 its modulus being the sine of the semi-angle between the focals. 



2nd. The rectification of the spherical ellipse is made to depend 

 on a complete elliptic function of the third order, whose parameter 

 is the same as in the preceding case, but whose modulus is the sine 

 of the angle between the planes of the elliptic base and of one of 

 the circular sections. 



The author then proceeds to establish a remarkable relation be- 

 tween the area of a given spherical ellipse and the length of the 

 spherical ellipse generated by the intersection of the supplemental 

 cone with the same sphere. 



He shows that if there are two concentric supplemental cones cut 

 by the surface of a concentric sphere, — 1st, the sum of their spherical 

 bases, together with twice their lateral surfaces, is equal to the sur- 

 face of the sphere ; 2nd, the difference of their spherical bases is 

 equal to twice the difference of their lateral surfaces. 



Hence, also, he deduces a remarkable theorem, viz. the sum of 

 the spherical bases of any cone whose principal angles are supple- 

 mental, cut by a sphere, together with twice the lateral surface of 

 the cone comprised within the sphere, is equal to the surface of the 

 sphere. 



The author then, alluding to some researches of Professor 

 MacCullagh and of the Rev. Charles Graves, Fellow of Trinity Col- 

 lege, Dublin, proceeds to give a simple elementary proof of a well- 

 known formula of rectification, and thence deduces some remark- 

 able properties of the tangent at that point of the ellipse, which is 

 termed by him the point of rational section. 



Assuming the properties of the plane ellipse, he proceeds to show 

 that a similar formula of rectification holds for any curve generated 

 by the intersection of a spherical surface with a concentric cone of 

 any order. He goes on to develope a series of properties of the 

 spherical ellipse, bearing a striking analogy, as indeed might have 

 been expected, to those of the plane curve. Thus he establishes a 

 point of rational section as in the plane ellipse, shows that the tan- 

 gent arc is at this point a minimum, and developes some other cu- 

 rious analogies. It is a simple consequence of his formula that the 

 spherical elliptic quadrant may be divided into two arcs whose dif- 

 ference shall be represented by an arc of a great circle. This 

 theorem, previously obtained by M. Catalan, is analogous to that of 

 Fagnani, which shows that the difference of two plane elliptic arcs 

 may be represented by a straight line. 



The author concludes by reducing the quadrature of the surface 

 of a cone of the second degree, bounded by a plane perpendicular 

 to the axis, to the determination of a complete elliptic function of 

 the second order. 



