424 Olbers' Essay on Comets. 



[For sin. (a + h) sin. c — sin. (a + c) sin b is, in general =: 

 sin. a COS. b sin. c — sin. a cos. c sin. b ■=. sin. a sin. (6 — c) . Tr.] 

 Dusejour finds a quadratic equation, not for ^', the curtate dis- 

 tance, but for the true distance, which he calls A'. But his 

 co-efficient for a'« becomes also iz 0, when the three places 

 of the comet are in a great circle. This co-efficient is equiva- 

 lent to sin. & COS. |3" co&. 0" sin. («" — a'") + sin. Q" cos. &f 

 cos. ^"' sin. (a'" — a') + sin. &'" cos. & cos. ^ sin.(*' — a"), 

 which, divided by cos. &' cos. /3" cos. /9'", becomes equal to S. 



§31. 

 It might also be shown that the two other co-efficients, in this 

 case, which is essentially the same with the supposition of a 

 rectilinear and equable motion, must both vanish: but we have 

 already sufficient evidence of the degree of utility of this me- 

 thod : and since three neighbouring observations of the comet 

 must always be very nearly in a great circle, the co-efficients 

 S, P, Q, which depend only on the curvature of the apparent 

 path, must always be very small, so that their values may be 

 materially altered by the errors of observation . When we add 

 to this consideration the want of perfexit accuracy in the deter- 

 mination of M and N, or of the proportions of the three cur- 

 tate distances, we shall find that this method is utterly useless 

 for neighbouring observations, aiid will in general afford a very 

 erroneous result. If, however, we had a sufficient number of 

 accurate observations, following each other at small distances, 

 the first, middle, and last of them being tolerably remote from 

 each other, so that we might determine M and N for them by 

 means of the intermediate ones, we might obtain something like 

 a solution of the problem from this method : and the most 

 readily where the apparent path of the comet deviated most 

 from a great circle ; which is most likely to happen when the 

 distances of the comet and the earth from the sun are very 

 different from each other, and when the comet is near the qua- 

 drature, or remote from the conjunction or opposition. But 

 after all, the calculation would be not a little tedious, and its 

 result too uncertain to be put in competition with those of other 

 methods of approximation. 



