180 Astronomical and Nautical Collections. 



t" sin {c"—a!) ^^ , , . , 



——. — —-^ -. = M ^ ; which expresses the proportion of the 



t sm (a, —c ) r r 



curtate distances in the first and third observations. 

 § 37. 



This mode of finding the proportion of the curtate distances is, 

 however, neither universally applicable, nor always the most 

 convenicnt.There is one case in which it is quite useless, that is, 

 where the apparent motion is nearly perpendicular to the ecliptic, 

 or the change of longitude very slow, and that of latitude con- 

 siderable : for in this case the angles c" — a' and «'" — c" 

 would be too small for the determination of M with sufficient 

 security. In another case it must be employed exclusively, 

 when comets are near their quadrature, and have very little mo- 

 tion, especially in latitude ; so that the following method is 

 rendered inconvenient. There is also another case, in which it 

 is particularly advantageous ; that is, when the intervals are 

 very small, or the observations not very accurate ; for in this 

 instance we may, without hesitation, employ the longitude a.' 

 of the second observation, instead of the corrected longitude c", 

 so as to supersede the calculation of § 35. This is equivalent 

 to the supposition, that the lines B b and D c?, § 34, are pa- 

 rallel to each other ; which will not be far from the truth when 

 the arcs a c and AC are small, and the lines b d, BD, still 

 smaller in proportion. In this case we may assume at once 

 M — <" sin ( a "— g') 

 t' sin [a!" — a." 



§ 38. 



In other cases it will be generally more convenient to employ 

 a plane of projection perpendicular to the ecliptic, and to the 

 middle position of the revolving radius belonging to the earth, 

 as Lambert has already done with advantage. If we then 



make "tang V = _.tang£_ ^ j"_ ^^"f/" ■ and tang 

 sin (A — «'^ sm (A — c ) 



^.„ _ i?^n^ & ^ ^l^g angles b, b", and b'" will be those 



sin (A" — cc") 



which the projections of the lines of direction will form with the 



tano" "v" tans' 8 i • 



earth's orbit. But smce ° ^ „ = — -— ^ -,v > tlie cal- 



sin(A — c ) sni (A — a ) 



