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SCIENCE 



[N. S. Vol. XLV. No. 1171 



misled to derive a parabola corresponding 

 to a solution other than the physical solu- 

 tion. By the method of the greatest com- 

 mon divisor Pieard has reduced the equa- 

 tions for general and parabolic orbits to a 

 linear equation giving the only possible 

 parabolic solution. 



The other point on whioli I desire 

 to dwell is that of the identification of 

 newly discovered planets or comets with 

 objects previously observed and for which 

 orbits are available. More than once it has 

 been found in the case of a newly discovered 

 comet that the inclination, node, and peri- 

 helion distance of the parabolic solution 

 resemble within the range of the solution 

 the corresponding elements of some former 

 parabolic comet. By introducing a period 

 corresponding to one or more revolutions 

 between the dates of the perihelia of the 

 two comets the original solution may be 

 turned into a conditioned solution based on 

 a definite period. In no case where definite 

 reasons for such procedure existed did this 

 experiment fail of proving the identity of 

 two comets, Thereby the two objects in- 

 stead of being different comets with para- 

 bolic orbits were recognized to represent a 

 single comet, moving in a definite ellipse. 

 It is quite probable that a proper study 

 of the existing comet lists may readily lead 

 to many identifications. Many pretty re- 

 sults might be cited in connection with the 

 various advantages to which I have referred 

 above as obtainable from a proper formu- 

 lation of Laplace's method. Undoubtedly 

 there are cases where Gauss's and Olbers's 

 methods would converge more rapidly than 

 my own formulation of Laplace's method, 

 but these are readily ascertained at the 

 outset. Orbits have been computed at 

 Berkeley for practically every comet since 

 the methods have been perfected, and so 

 far every ease has readily yielded to a solu- 

 tion. 



In recent times notable memoirs have 

 been written on orbit theory by Harzer, 

 Charlier, and Moulton. I have not as yet 

 had an opportunity to study Harzer 's new 

 geometrical methods with respect to their 

 practical value. With the claims made in 

 Moulton 's memoir on the ' ' Theory of Deter- 

 mining Orbits," published in the Astro- 

 nomical Journal in 1914, I can not, un- 

 fortunately, find myself entirely in accord. 

 The object of the memoir is set forth to be, 

 on the one hand, to clarify the problem 

 mathematically, and, on the other, to define 

 the extent of the indeterminateness. In 

 spite of the noted mathematical skill of 

 Moulton it appears that although his forms 

 have the merit of symmetry, his treatment 

 of the problem which involves determinants 

 of the ninth order, though readily reduced, 

 offers no advantages over the simplifications 

 arising from earlier combinations of geo- 

 metrical amd dynamical relations. To his 

 misconception of the practise of the com- 

 puters at Berkeley, with reference to the 

 interpretation of the accuracy of their re- 

 sults, I have already referred. These mis- 

 conceptions apply also to the significance 

 of a number of theoretical and practical 

 points, particularly to his interpretation of 

 the vanishing of the chief determinant. 

 Quite contrary to his statement in his 

 "Celestial Mechanics" that in general the 

 expressions for p and p' become indeter- 

 minate when the determinant referred to 

 is zero and that they are poorly determined 

 when it is small, it may be shown that the 

 orbit is in general well determined when the 

 determinant is definitely zero or definitely 

 small, and that the determinateness of the 

 solution does not depend so much on the 

 numerical value of the determinant, but 

 upon the accuracy with which this numer- 

 ical value can be found. Thus a large 

 range of solution may result for a compara- 

 tively large value of the determinant, if 



