36 Dr. Booth on the 'Rectification and 



XXXIV. Let 2 5' be the angle between the asymptots of 



the spherical hyperbola passing through the extremity of the 



sin /3" 

 arc measured from the major axe, then, as before, sin 5' = -: — r., 



J sm a" 



a" and |3" being the principal semiangles of the cone of which 



this hyperbola is a section. 



It may with little trouble be proved that 



.,, tan B' . „ . , 



tan /3" = r-»j sm «" = cos A'. 



"^ tan A' 



Substituting for the functions of A' and B' in these equations 

 their values given in (62.), we find, after some obvious reduc- 

 tions, 



tan^ /3 sin^ A 



sin^ fl' = 



cos2fl' = 



tan^ « cos^ \ + tan^ /3 sin^ x' 

 tan^ a cos^ \ 



tan^ « cos'"' A. + tan^ /3 sin^ A ' 



hence tan 9' = ^^ tan X (69.) 



tan a ^ 



Multiplying {Q6.) (69.) together, we obtain 



tan9tane' = ^, (70.) 



sm a 



a result independent of A, and in strict conformity with (39.). 



In the polar equation of the ellipse, substituting the values 



of sin 9' cos 9' given above, p' being the corresponding radius 



vector, we obtain the resulting equation 



. o # tan^ a cos^ A 4- tan^ /3 sin^ A ,^, v 



sm-'p'ss — 2 27""; 2 ,. • o ^ ; • • (71.) 



^ sec'acos^A + sec^psm^A ^ ' 



hence sin p sin p' = sin a sin /3 (72.) 



a result also independent of A. 



We have thusp' = w; or the semidiameter of the ellipse along 

 the asymptot of this hyperbola is equal to the perpendicular 

 from the centre on the tangent to the ellipse drawn through 

 the point R of intersection of the ellipse and hyperbola. See 

 fig. 4. 



XXXV. Let r and r' be the semidiameters of the spherical 

 ellipse passing through the points w, jm., in which the ordinates 

 of the extremities of the elliptic arcs being produced meet the 

 circle on the major axe ; let •& and ^' be the angles which r and 



r' make with the major axe (fig. 4), then ^ = -^ A, and the 



value of d' may be thus found; calling H the spherical coor- 

 dinate of the point j«, on the circle, 



tan^^ tan^H . , . , 



:: — 3 — I- -: — s — = 1 in the circle, 

 tan^a tan^a 



