B2 Dr. Booth on the Rectification and 



These lines are termed focals, and the points in which they 

 meet the surface of the spherical ellipse, are analogous to the 

 foci of the plane ellipse. 



Let e be the eccentricity of the plane elliptic base of the 

 cone, then 



„ a^ — b^ tan*a— tan^/S sin^a— sin^^ .„. 



a* tan'^a sm'^acos-'jS 



1 , / V ' a sin^ « — sln^ /3 



and by (1 1 .) sm^ t = ^^ — ^• 



^ ^ ' cos^/3 



Introducing the relations established in (11.) and (12.) into 



(10.) J we find 



^^_tnn^ /*y^ r \ 1 (j3^j 



2 tan« c/o ^L(l_e2sin2^)>v/l-sin2£sin2<pJ 



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

 meter is of the circular form, as might be easily shown. This 

 appears to be the simplest shape to which the quadrature 

 of the spherical ellipse can be reduced, the parameter and 

 modulus being the eccentricities of the plane and spherical 

 ellipse respectively^ 



VIII. To find the length of an arc of the spherical ellipse. 



In this case it will be found much simpler to integrate 

 the equation (3.) with respect to f^ instead of co, the independ- 

 ent variable in the last problem ; for this purpose, then, solving 

 equation (5.), we find 



. o sin^ /3 fsin^ a — sin^ p~\ 



sm^w- -^<-^ ^r • • • (14-) 



o sin^ a fsin^ p — sin^ /3" 



cos-' « i= -^-g- ^ -^-o^ ^-2^ r* • • • (iM 



sm^ p Lsm^ a — sm* /3 J ^ ' 



Differentiating (14.) with respect to w and p, 

 d(a — sin^ a sin^ j3 cos p 



sin CO cos W -7- = . 3 . . o ^-2fl\* 



a/> sm"* p (sm^ a — sm-* /3) 



Dividing this equation by the square root of the product of 

 (14.) and (15.), we obtain 



/7 CO — sin a sin Q cos s 



_ — ! ' — ^1 fi ^ 



df siiip -/sin^a — sin^p Vsin^p — sin^/3' *' 



Substituting this value of ^- in the formula (3.) for the rec- 



M.A., Fellow of Trinity College, Dublin, who has enriched his version with 

 very valuable notes, and an appendix containing amongst other original 

 matter a new theory of rectangular spherical coordinates, which is likely 

 to become a powerful instrument of investigation in researches of this 

 nature. 



