traced upon the Surface of the Sphere. 279 



(cos e, cos e a cos /? /3, ; ) sin /3, + (cos e,, cos e, cos ft,, {3,) cos /3 , 

 cos ri t^ ^i . -_'--L'_- ___ -^-^ ", . , ( y. i 



sin /3,, /3, V cos 2 e, 2 cos e, cos e a cos /? /3, + cos z e,, 



~ ** cos 2 e, 2 cose, cos e, cos/3,, /3, +cos 2 e,, 



The expressions in (7, 8, 9), though perfectly symmetrical, are not the 

 simplest that can be given to those quantities. For, taking the numerator 

 of (7), and expanding the cosines, and multiplying out, 



(cos e, cos e,, cos /3 /; /3,) cos /3 / + (cos e,, cos e, cos /3,, /?,) cos (3, 

 =cose / {cos/5 / -cos 2 /3 // cosj8 / -cos/3 // sin/3 // sin 1 S / }+cos // {cos/? // -cos^ // cos*/3 / -cos^ / sin/? / sin/3, / j 

 - cos e, {sin Z /3 X/ cos /3, t sin /? sin /3 7 cos /3 /x } + cos e w {sin ^ cos^ /( sin^, sin/3, cosjS,} 

 = {cos e ( sin /3 /7 cos e /x sin ft,} {sin /? cos (3, sin /3, cos /?} 

 = {cos6 / sin/9 // cos ^ sin ^} sin jS,, ^, ............................... , ................ (H) 



In like manner, for the denominator of equation (7), we have 

 cose, (sin/3, - sin 2 /?/, sin/3,-cos/3,,sin/9,,cos/3,) +cose // (sin/3, / -sin 2 /3,sin/3, / - sin/3, cos/3, cos/3,,) 



cos e, (sin /3, cos 2 /3,, cos /?/ cos ^,, sin /3,,) + cos e,, (sin /3,, cos */3, sin j3, cos /3, cos /3,,) 



(cos e, cos (3,, cos e,, cos /3,,) sin /3,, /3, ...... .................................... (12.) 



The equations (7, 8, 9) then become, respectively, 



tan/c = COSf - sin ^- C S ^ sin ^... ...(13.) 



cos e, cos p,, cos e,, cos p, 



cos e, sin /3,, cos e,, sin /3, 

 sin /c = q= = ............................. (14) 



V cos 2 e, 2 cos e, cos e,, cos p,, p, + cos 2 e,, 



cos e, cos /3,, cos e,, cos /3, 



(15.) 



_ - - 

 V cos*e, 2cos e, cos e,, cos/3,, p x + cos 2 e,, 



Now, in the general equation of the circle, (I.) of t-his paper, put the 

 values of tan X, cos K, and sin K, then there will result the final equation 



cot<= gfcos^^ose^cos/S,, cos e,, cos /3,) + sin 6 (cose, sin /3,, cose,,sin/3 / )}..(16.) 



Cor, If e x and e /7 be right angles, we shall have cot <p = Q, whatever Q 

 may be, as we otherwise know it should be, the circle sought being then 



the equator. 



N n 2 



