Quadrature of the Spherical Ellipse. 

 Now this equation may be written in the form 

 ^-3— (1 — sin^ «) + ^-2-^ (1 - sm2 ^) = 



21 



sin^p 



sin^p 



or 



cos'' «> sm-' CO 



sm*a sin^/3 sm-^p 



which is the equation of the spherical ellipse in another form. 

 V. Dividing (4.) by (5.), and reducing, there results 



cos p = cos a. 



^/ 



/, sin^ a ^ g 



, , tan^a^ 2 

 tan-^p 



(6.) 



sin^ /3 



Substituting this value of cos p in (2.), integrating, and putting 

 A for the area of a quadrant of the spherical ellipse (for, as 

 the surface of this spherical ellipse evidently consists of four 

 symmetrical quadrants, the length or area of one quadrant is 

 one-fourth of the length, or of the area of the whole). 



A = — — cos a I di 



2 c/q 



v/ 



, , tan^a 2 

 tan^p 



^/ 



1 -f . o ^ tan^ CO 



sin«/3 



(7.) 



VI. Now this definite integral is an elliptic function of the 

 third order, as may be thus shown. Assume 



tanlB^ 



tan ip, . . . . . 



tan CO = 



then 



doi 



d<p 



tan a 



tan « tan /3 



(8.) 

 (9.) 



tan^ a cos^ (p + tan^ |3 sin^ <p* 

 Introducing the relations established in (8.) and (9.) into (7.), 

 the resulting equation becomes 



tan 

 tan 



- cos « / , . — 7-7, r 



« *y Q "Y \ J i _ / sin^a — sin 



LI \ sin' «6 cos^ 



?). 



sin^^ 



-sin^^ 



VII. Let two right lines be drawn from the vertex of the 

 cone in the plane of x ^, or in the plane of the principal atigle 

 2 «, making equal angles e with the real axe of the cone, so 

 that 



cos s = -: (11«)* 



cos p ^ ^ 



• The most accessible treatise to which I can refer the reader desirous 

 of information on the subject of cones and spherical conies, is a translation 

 of two Memoirs of Chasles, lately published by the Rev. Charles Graves, 



