June 8, 1917] 



SCIENCE 



581 



been extremely simplified. A conditioned 

 solution may be made on practically the 

 same plan as a general solution. Criteria 

 have been introduced to distinguish be- 

 tween the feasibility of solution with or 

 without assumption regarding the eccen- 

 tricity. Provision has been made for pass- 

 ing from one class of orbit to another in 

 the course of the computation without re- 

 peating the solution. Numerical criteria 

 have been set up to distinguish the phys- 

 ical from the mathematical solutions in the 

 ease of three roots which may occur in the 

 parabolic method. A method has been pro- 

 vided for completely eliminating the para- 

 lax, as has been done in the case of Planet 

 MT by Dr. E. S. liaynes, since in this case 

 the possibility of a solution rested on such 

 elimination. The various approximations 

 for the solution of distances is avoided. In 

 a general solution the distances are taken 

 from a table. The accuracy attainable in 

 each ease can be ascertained in advance 

 and the range of solution definitely deter- 

 mined. Series coi-responding to the ratios 

 of the triangles, which do not enter, how- 

 ever, in the original solution but only later 

 after the distances have been determined, 

 have been replaced by closed expressions 

 which avoid slow convergence or divergence 

 in case of comet orbits observed near peri- 

 helion and at a moderate distance from the 

 sun. The whole cycle of hypotheses and 

 approximations of the older methods and 

 all initial inaccuracies are taken up by a 

 method of differential correction. In the 

 case of highly disturbed satellites, such as 

 the ninth satellite of Jupiter, the orbit solu- 

 tion has been made possible by extending 

 the formula so as to take account of the 

 perturbations in the first approximation. 

 Closed expressions in the differential cor- 

 rection of an orbit now make it possible to 

 apply the method to any and all conditions, 

 particularly to arcs of any length. 



It is not possible to dwell further on 

 these advantages, yet reference may be 

 made to some important results which 

 make unnecessary extensive investigations 

 hitherto in use. In the case of comet 

 1910a, which was discovered near peri- 

 helion and at a moderate distance from the 

 sun, a variety of preliminary orbits were 

 derived by various computers. Through 

 the work of Oppolzer, Charlier, and myself, 

 it was already known that cases of three 

 mathematical solutions might be possible. 

 Tscherny classified all the different pre- 

 liminary orbits that had been derived for 

 this comet and showed that clearly they 

 represented three groups, each group repre- 

 senting a range of solution clustering about 

 a mathematical solution. Each computer 

 had produced a perfectly legitimate orbit 

 within the errors of observation, none recog- 

 nizing that his orbit was one belonging to 

 one of the three ranges, or that multiple 

 solutions existed. In my short methods 

 simple criteria are given for determining 

 the existence of three mathematical solu- 

 tions of the equation of the sixth degree for 

 a parabola. As soon as observations be- 

 came available the method was applied by 

 Miss Levy and three distinct values of the 

 geocentric distance at the middle date and 

 the range of each were obtained. A simple 

 consideration leads to the elimination of 

 the two fictitious parabolic solutions, as 

 there can be at most two general solutions, 

 also either one or three parabolic solu- 

 tions, and as only one parabolic solution can 

 agree Avith a general solution. By this 

 process the physical solution was at once 

 determined. The two general solutions 

 corresponding to the problem are readily 

 taken from the table so that all five roots, 

 two general and three parabolic, are avail- 

 able simultaneously and with little effort. 

 Therefore, there really exists no reason 

 why hereafter a computer should ever be 



