June 8, 1917] 



SCIENCE 



583 



that value has a large percentage error. 

 These conditions have been partially set 

 forth in Buchholz's "KHinkerfues Theoret- 

 isehe Astronomie, " but reference has been 

 made by Moulton only to the first and 

 initial draft of the methods as published 

 in 1902. It is, of course, not my intention 

 at this time to undertake a detailed anal- 

 ysis of Moulton 's memoir. This must be 

 deferred to some more appropriate time and 

 must be done in more explicit form. With 

 reference to the formulation finally adopted 

 by Moulton it may readily be shown that 

 it reverts to Gauss's method. 



The most notable and classic contribu- 

 tions to the orbit theory in recent years 

 have been made by Charlier. In a num- 

 ber of memoirs he has set forth the funda- 

 mental principles of the problem and has 

 thrown much light on the subject with 

 reference to many details, but his most im- 

 portant contribution is the resumption of 

 Lagrange's incomplete analytical solution, 

 a pure analytical solution in series, which 

 admits of determination of the higher 

 terms by direct computation without in- 

 volving successive approximations of any 

 kind nor requiring an improvement of the 

 lower terms. His theoretical developments 

 hold out the highest promise of successfully 

 conquering the problem in practise without 

 the complications existing in the older meth- 

 ods. But at present serious practical diffi- 

 culties still exist, chief among which is that 

 the series involved become extremely com- 

 plicated wdien a high degree of accuracy 

 is required, and that the method is subject 

 to several of the limitations of the older 

 methods. If these complications and 

 limitations can be removed the method will 

 be the best in existence. One of the 

 chief limitations affecting practise con- 

 sists in the fact that it is a general method 

 and that it therefore may lead to orbits 

 within the range of solution which are not 



acceptable from experience. It has been 

 applied to the computation of several 

 planet and comet orbits by Charlier and his 

 associates. In the case of Comet e 1906 the 

 resulting orbit is an hyperbola with an eccen- 

 tricity 1.46. This seems to represent a solu- 

 tion near the upper edge of the range. A 

 parabolic solution has been produced from 

 the observations without difficulty in the 

 first approximation by my formulation of 

 Laplace's method by Miss Levy. In the 

 case of another comet the elements are 

 slightly hyperbolic; in the case of planet 

 (702) the orbit deduced by the Charlier- 

 Legrange method from an arc of two 

 months gives an angle of eccentricity 

 differing by nearly four degrees from the 

 corresponding angle deduced near the 

 upper possible limit by Miss Levy. Under 

 these circumstances it would be diffi- 

 cult to decide where to stop in the compu- 

 tation of the terms of his series in relation 

 to the possible range of solution. 



From the somewhat disconnected and in- 

 complete observations which I have just 

 made on the methods of determining orbits 

 it is seen that the interest of investigators 

 is directed along two distinct lines, purely 

 mathematical and practical. A proper 

 adjustment between the two is required 

 by the demands of astronomical science. 

 In this connection and in conclusion I 

 may make reference to the possibility of 

 determining the orbit of a highly dis- 

 turbed satellite from a limited number of 

 observations on the basis of Laplaeean 

 principles. It is not necessary to await 

 the evaluation of all the 18 integrals of 

 the problem of three bodies for the 

 purpose of setting up a satisfactory orbit 

 method. Laplace 's method for the two-body 

 ease is not based on Newton's integrals, 

 but by introducing numerical values for 

 the geocentric velocities and accelerations 

 in a and 8 the differential equations are 



