sin/c 



= [ 



292 Mr DAVIES on the Equations of Loci 



Subtract (3) from (2), and (4) from (3) ; then we have 



cot \ (cos a cos a,) = sin a, cos (ft, K) sin a cos ((3 K) (5.) 



c . - 



cot \ (cos a cos a,,) = sin a a cos (ft,, K) sin a cos (p K) ...(6.) 



i< . t i '. 



Expand these, and arrange them according to cos K and sin K. Then 



cot X (cos a cos a,) = (sin a, cos ft, sin a cos (3) cos K + (sin a, sin ft, sin a sin ft) sin K (7. ) 



cot \ (cos a cos a a ) = (sin a,, cos /? sin a cos /5) cos K + (sin a a sin /? sin a sin /3) sin K (8. ) 



Resolving these (7, 8) with respect to sin K and cos K, we have 

 cos a (sina // cos/3 << '-~ sin a, cos ft,) + cos a, (sin a cos/3 sin a a cos/?,,) + cosa,, (sin a, cos/3, sin a cos (3) 



cos/c 



=-[ 



sin a, sin a,, sin ft, ft,, + sin a sin a,, sin /3 ; , ft + sin a sin a, sin ft ft, J 



cosa (sin a,, ship*,, sin a, sin ft,) + cos a, (sin a sin/3 sin a,, ship*,,) + cosa^ (sin a, sin p\ sin a sin ft) 



IcotX 



(9.) 



sin a, sin a,, sin/3, ft,, + sin a sin a,, sin /? /3 + sina sin a, sin/3 ft, 

 Write these, for the present, in the following form, 



"IcotX (10.) 



n 



m 



sin K"= cot A, and cos K = cot X. 

 P P 



Then, since sin 2 + cos 2 K z: 1, we have 



t % 



w 2 + n 2 

 tan 'A = j , and hence 



P* 



sin X = 



cos X = 



.(11.) 



Hence also we have 



n n 



sm K = cot X = -= 

 P 



m 



+ '> 

 m 



, and 



COS K COt X == =^__ 



P tjrr? + rf 



.(12.) 



Inserting these several values of the functions of K and ^ in either of the 

 equations (2, 3, 4), we should get values of cosp; but in these cases, the 



