determining the Orbits of Comets. 15 



[f> r"] q (tan 9 sin (a - 8') - tan 8 sin (a' - 6')) 

 + [rV] R tan 8' sin (6' — 6) 

 + [r r 7 ] $" (tan V sin («"- 6') - tan 8" sin (a' - 6')) 



- [r /] R"tan 8' sin (6" - 6') = 0, 

 whence 

 * — [r'*"] tan 8 ' sin (a — 6' ) - tan 8 sin (a! — B ! ) 

 P Z [r r 1 ] ' tan 8" sin («' - 6') - tan ? sin {^ f ^ r e') ' P 

 tanj; [r r r"] R sin (6' - 0) - [r tj JR" sin (ylflQ 

 + [r r] ' tan 8" sin (a' - 6') - tan 8' sin (*" — e) ' 

 and introducing the analogous symbols 



[R' R"] = R' R" sin (6" - e') 

 [R R"] = R R" sin (e» - 6) 

 [R R'] =RR' sin (e' - 6), 



the second term of the right hand side of the equation may 

 be written 



/ [tV] [R'R"] \ R tan 8' sin (6' - 6) 



+ \ [rr'] [R R'] / ' tan 8" sin (<*'-#') -tan 8' sin («"- e')" 

 We have, therefore, assuming 



M , _ tan 8' sin (a — &') — tan 8 sin (a! — O') 

 ~ tan 8" sin («' — 6') — tan 8' sin («" - 0) ' 

 O^ M"- tan 8' sin (6' -6) 



,,_ [rV'] 



Mp+ V[rr'] [RR']/ M K > 



an expression as yet perfectly rigorous. 



The areas of the triangles \r r'],&c. 9 are in themselves, when 

 the intervals of time are small, little different from the sectors 

 of the parabola to which they belong, and as little will the 

 ratio of two adjoining sectors be much different from that 

 of the two areas of the corresponding triangles, as at any rate 

 each is smaller than its corresponding sector. For the orbit 

 of the earth this will be equally the case, and the more so as it 

 approximates so nearly to a circle, and as there is consequently 

 a case in which the equality of these ratios may be almost con- 

 sidered as rigorously true, viz. when the intervals are equal. 

 But the sectors are proportional to the times. We may there- 

 fore suppose as very nearly true, 



,,sv |>y']_[R'R"] _;"-;' 



S ' [rr'] ~~ [RR'] " t' -t' 



By these equations the last term in (14) becomes quite 

 evanescent, and we obtain 



