June 8, 1917] 



SCIENCE 



577 



fues, by "Watson in this country, by Op- 

 polzer, by Buchholz, by Tisserand, and 

 others, and are in use to the present day in 

 accordance with the various formulations, 

 but unfortunately without being duly ap- 

 preciated in all cases by computers with 

 reference to their numerical significance, 

 that is, with reference to the validity of the 

 results which they produce as conditioned 

 by partial indeterminateness or range of 

 practical solutions. Furthermore, to cite 

 from my paper on "Preliminary Statistics 

 on the Eccentricities of Comet Orbits": 



Ever since the first computation of a comet orbit 

 was made, it has been customary to derive a para- 

 bola as a first approximation, and to attempt a 

 more general solution only if the deviations of the 

 observed positions from the most probable para- 

 bola were in excess of the probable errors of ob- 

 servation. This custom has become so thoroughly 

 fixed in astronomy that even now it would be con- 

 sidered absolutely unwarranted to suspect a comet 

 of moving in an ellipse if by a little stretching of 

 the probable limits of observational error a para- 

 bola could be found to represent the observed posi- 

 tions. A prejudice has always existed and exists 

 now in favor of the parabola for comets. This 

 prejudice is largely due to the many published 

 parabolic comet orbits. A further reason lies in 

 the fact that the first geometrical and analytical 

 methods for solving a comet orbit were parabolic. 

 The solution of an elliptic orbit was originally pos- 

 sible only in cases like HaUey's comet, in which 

 more than one appearance had been observed so 

 that one of the unknowns, the period, became 

 known. 



The procedures in the older methods for 

 the derivation of a parabolic and the deriva- 

 tion of a general solution are so different 

 that when it is recognized that a parabolic 

 orbit or conditioned solution is not possible, 

 a fact which does not reveal itself until 

 after many fruitless attempts at a parabolic 

 solution have been made, it is necessary to 

 discard most of the previous numerical 

 work and to start anew with the formula 

 for a general solution. An illustration of 

 the labor involved in this antiquated proc- 

 ess is furnished by the published work of 



one of the leading European astronomers on 

 the preliminary orbit of comet 1892 II. 

 (Holmes). Three observations at equal 

 intervals of four days were available in this 

 ease. The computer attempted a parabola 

 and, finding that he could do nothing with 

 the ratios of the triangles in improving his 

 orbit, finally resorted to an arbitrary varia- 

 tion of M referred to in Olbers's method as 

 the ratio of the third to the first geocentric 

 distance, thus producing four different 

 parabolic orbits with ephemerides from 

 which to choose, none of which admittedly 

 represented the given observations. Only 

 later and still greater discrepancies between 

 observation and the predicted path lead the 

 computer to resume the computation with- 

 out hypothesis regarding the eccentricity. 

 In due course of time this general solution 

 yielded a short period ellipse. Here is a 

 bit of practise still in use among many 

 computers which, if applied in the business 

 world, would involve an enormous cost of 

 operation. Mr. Shane, one of my students, 

 applied my formulation of the Laplacean 

 method to this case and obtained the true 

 ellipse in the first approximation or by a 

 direct solution without difficulty in a few 

 hours. 



Nor is it always safe to assume the na- 

 ture of the object and the character of its 

 orbit from its appearance. Thus some 

 comets are of a star-like appearance when 

 discovered and can not be distinguished 

 from asteroids, and to prejudice the char- 

 acter of the orbit at the outset may lead to 

 unnecessary complications. It is not im- 

 probable that some of the short periodic 

 orbits published for supposed minor planets 

 which have become lost are really very 

 eccentric orbits of comets which would ac- 

 count for their failing to be reobserved in 

 their predicted places. Just how many of 

 the published orbits of the hundreds of 

 planets and comets are entirely reliable is 



