Olbers' JSssoy on Comets. 417 



the earth, and even the elements and nature of the orbit, from 

 three observations at short intervals of time. The equations 

 deduced from this supposition, retaining the notation already 

 adopted, would be t : t" = (x' — x"^) : {xf' — x'") = (y' — y") : 

 (y" — y"') = (^' — O : (2" — «'"), whence ^ ', e", and ^" might 

 be deduced by means of linear equations only ; and since these 

 values of ^' and ^"' would be independent of the motion in a 

 parabola, we might obtain from them, if they were perfectly 

 accurate, by comparing them with the whole time intervening, 

 not only the situation and dimensions, but also the nature of 

 the conic section in which the comet revolves. 



§ 22. 

 There is, however, a case in which the problem again becomes 

 undetermined, even when the lines are not in a single plane. 

 If they are all parallel, no right line can cut them all ; and in 

 other cases, there is only a single line for each point of one line 

 that can cut the other two, but it may happen that all such lines 

 must necessarily be cut in the same proportion ; and this will 

 occur when the points to which the three lines tend, speaking 

 astronomically, are found in a great circle, or geometrically, 

 when they are parallel to three lines lying in any one plane. 

 Such must be the case whenever more than one line is divided 

 in the same proportion by the three which it intersects. Hence 

 it follows, that if the portion of the earth's orbit, described in 

 the given interval, were a straight line, and the earth's velocity 

 equable, this line would be divided by the line of direction in 

 the same proportion with the supposed portion of the comet*s 

 orbit ; so that the relative position of these lines, with respect 

 to the lines of direction, would remain completely undetermined, 

 and Bououer's equation would lead to no conclusion whatever. 

 The conclusion obtained from it must therefore depend on the 

 deviation of the earth's orbit from a straight line, and on the 

 change of its velocity, whilst the curvature of the comet's orbit, 

 which is often considerably greater, is totally neglected. In 

 fact, if we suppose the earth's motion equable and rectilinear, 

 we shall have the equations i : t" ■=. (R' cos. A' — R" cos. A") : 

 (R" cos. A" — R" cos. A") = R' sin. A'— R" sin. A") : (R" sin; 



