Quadrature of the Spherical Ellipse. 3S 



tification of the arc of a spherical curve, the resulting equation 

 becomes 



/»Pi , r sinp v^cos^p — cos^acos^/3 "1 ,,^. 



arc = / dp , I . \ — , . ^=rr^ , . (17.) 



•^^ po L ^sin « - sin* /) ^^sin^ ^ - sin^ ^ J ^ 



the arc being measured from the minor axis towards the major. 

 IX. Let s be the arc of a spherical quadrant, then 



no. p sin p -v/cos^ p — cos^ a cos^ /3 "~1 / , % 

 s= / c?o , ^ -X- / . • (18.) 



-J e- ^ \_ V%\n^ « - sm2 p Vsm* p - sin^ /Sj 



This is a complete elliptic function of the third order, which 

 may be reduced to the usual form in the following manner. 



Assume g sin^ « cos^ (p -f- sin^ ;3 sin^ ip 



^°® ^ ~ tan2 a cos^ ^ + tan^ .3 sin* <p' ' ' ' ^^^'^ 

 the limits of integration being changed from a and /3 to 

 and — -, or (changing as well the order of integration as the 



sign) from — toO. Differentiating (19.) and introducing the 

 relations assumed in it into (18.), there results the equation 



tan/3 



5 = - — ^ sm 



tana 





X. Let y denote the angle which the plane of one of the 

 circular sections of the cone makes with the plane elliptic base, 

 then it may be shown with little difficulty that 



sin/3 .^, . 



cosy = -^— ^; (21.) 



' sma ^ ' 



or sin'^ y = r—^ -. Introducing this relation into (20.), 



TT 



tan/3 . . /»"2 , r 1 n ^ s 



5= — -sm Bf d<p\ . , (22.) 



tana \/o ^L(l - e2sin2(p) i/l - sin^y sm^f J '^ ^ 



a complete elliptic function of the third order, whose para- 

 meter is also of the circular form. 



XL Let u' and /3' be the principal semiangles of the sup- 

 plemental cone*^ and 5' the length of a quadrant of the spheri- 

 cal ellipse, the curve of intersection of this cone with the sphere, 

 then 



* A cone is said to be supplemental to another when then* principal 

 armies are supplemental. 



