Astronomical and Nautical Collections. 



179 



CD ; so that the lines AC, All, and AE, must always be in 

 one plane. — Tr.] 



§ 36. 

 Since, according to our supposition, the chord of the orbit 

 of the comet, and that of the earth's orbit, are divided by the 

 lines of direction a A, d D, c C, in proportion to the times, the 

 same proportion must also hold good for all orthographical pro- 

 jections of these chords and lines of direction. Supposing now 

 CDA to represent the chord of the orbit of the comet projected 

 on the ecliptic, a c d, as before, the chord of the earth's orbit, 

 and a A, dD, c C, lines determined in their angular directions 

 by the longitudes »', c", and «'" respectively : 



we have then CO : AM 

 c d 



CD 



AD 



and c O : 



sin COD • sin DMA 



but since c d : da ~ CD 



a M = . 



sm COD sin DMA 



AD = ^' : t', and C c =r CO + c O, and A a = AM + a M, 

 t' t" 



we have A a : C c = 



Now DMA is the 



sin DMA sin COD 

 difference of the first and second longitudes, c"— a' ; and COD, 

 that of the second and third, a" — c" ; and A a and C c are 

 the curtate distances of the comet from the earth in the first 

 and third observation, which we have before called §' and ^"' ; 

 consequently, 



and g'" = 5'. 



e • ? = 



sin (c" — a') sin(a"'— c") 

 N 2 



