Astronomical ami Nautical Collections. 181 



dilation of the values of c' and y" becomes unnecessary. [For 

 in this case the projection of the revolving radius of the comet 

 will coincide with that of the line of the direction m the second 

 observation, the same point representing both the earth and 

 the sun.] If we now call the projected distance in the first ob- 

 servation J', and in the third N ^, we shall have, since the pro- 

 jected chords are here also divided in the proportion of the 



.• ,T t" sin Cb" — b') ^^ 

 times, N = __ — 1-- jf- . Now 



l! &m(b"' — b") 



, ^ cos b' J ,„ M /_ N ^ cos b'" ,, 



P — , and e = Me = ; consequently 



sin(A"— a'j sin(A"-«"') ^ *' 



^^ _ cos b"' sin {k"-o<.') %m{b" -b') t" _ 

 "" cos b' sin (A"—*'") sin {b"'—b") t' 



sin (A" — a,') (tang 6" — tang b') t' 



sin (A" — a.'") (tang b"' — tang 6") t' 

 (tang 0' sin (A"— a') — tang <S' sin [A"—ac"] ) t" ' 

 (tang ^ " sin (A"— «")— tang 0' sin [A" — a'"] ) t' 

 a very convenient expression for M ; which, however, may be 

 rendered more immediately applicable to calculation in the form 



^^ (msin(A"-.-)-tangg;)r^ making ,« = _l^!lgl—. 

 (tang — ?« sin [A" — «'"] ) t' sin (A " — a.") 



[Note communicated by the Atithor. 



It must here be observed, that the two expressions M = 

 sin (c" — g') t" ^^^ -^ _ (m sin (A" — a') — ta ng 0) t' 

 sin [a!" —c") t (tf-rig 0" — m sin (A"— a!")t" 



identical, and may be derived from each other. For since 



wt = -: -77 r is the tang-ent of the anarle formed with the 



sin (A -a) 



ecliptic, by a great circle drawn through the place of the 

 comet and of the sun in the middle observation, we obtain, 

 by means of a well-known property of great circles, already 

 mentioned in § 30, the two equations 



tang y sin (a'"— a')— tang sin {a"'—c") -0" sin (c"— a')—0 

 tang y sin («'"— a') — m sin (A" — a') sin (a'" — c") — m sin 

 (A" — a'") sin (c"— a') = 



sin (c" — a') m sin (A" — a') — tang -, 



consequently ^^^.,_^.>^ - tang i3"' - m sin (A'-a'")'"' 



