VOL. LXII.] PHILOSOPHICAL TRANSACTIONS. 351 



to AB, in tlie triangle, or as the cosine of the spherical angle abd, to the cosine 

 of the spherical angle bad. 



If IK. be the situation of the horizon, when the solstitial point is ascending, in 

 the quadrantal triangle aik, the cosine of kic is to the radius, as the cosine of 

 iKA (= DBA) to the cosine of iak. Therefore the cosine of bdc, when d is the 

 point of longest ascension, is equal to the tangent of half the complement of the 

 angle which the ecliptic makes with the horizon, when the solstitial point is 

 ascending. 



But the sine of the angle composed of dab and twice abd, must be less than 

 3 times the sine of the angle bad. In the spherical triangle abd, the angles 

 bad, abd together exceed the external angle bdc. Therefore, in the 3d corol. 

 of the lemma, let the angle ban be equal to the sum of the spherical angles bad, 

 ABD : but here, an is to ab as the cosine of the spherical angle abd to the co- 

 sine of bad ; and an is also to ab as the sine of abn to the sine of anb, that is, 

 as the cosine of bap to the cosine of nap ; consequently, since the angle ban is 

 equal to the sum of the spherical angles bad, abd, the angle nap is equal to the 

 spherical angle bad, and the angle bap equal to the spherical angle abd ; but the 

 sine of the angle composed of nap and twice pab is less than three times the 

 sine of nap : therefore the sine of the angle composed of the spherical angle bad 

 and 2ABD will be less than three times the sine of the angle bad ; otherwise no 

 such triangle dba, as is here required, can take place, but the point a will be 

 the point of longest ascension. 



If the sine of the angle a be greater than 4- of the radius, the point a can 

 never be the point of longest ascension; but when the sine of this angle is less, 

 the angle compounded of bad and twice abd, may be greater or less than a 

 quadrant ; and therefore the magnitude of the angle abd, that a be the point of 

 longest ascension, is confined within 2 limits, of which the double of one added 

 to the angle a, as much exceeds a quadrant, as the double of the other added to 

 that angle falls short of it ; therefore double the sum of those two angles, toge- 

 ther with twice a, makes a semicircle; and the single sum of those two angles 

 added to a makes a quadrant. 



Prob. II. — To find when the ^rc of the Ecliptic Diners Most from its Oblique 



Ascension. 



Analysis. — If (fig. 8) bd be the situation of the horizon, when cd differs 

 most from cb, as before, the ultimate ratio of be to df, will be compounded of 

 the ratio of the radius to the sine of dg (or the cosine of db) and of the ratio of 

 the sine of cb to the sine of cd : but when cd differs most from cb, be and dp 

 are ultimately equal ; therefore then the cosine of bd is to the radius as the sine 

 of cb to the sine of cd. 



