572 



SCIENCE 



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



body problem. In fact these orbit methods 

 may be characterized as an evaluation of 

 the numerical values of the constants or 

 elements from given positions on the basis 

 of the integrals found by Newton. It might 

 be supposed that the mere evaluation of the 

 numerical values of the constants of inte- 

 gration in a given case when the form of 

 the integrals is kno^vn ought not to involve 

 any considerable difficulties. But the solu- 

 tion of the unknown elements from the 

 given equations of condition leads to very 

 complicated expressions which can be 

 solved only by successive approximations. 

 This unfavorable condition arises from the 

 occurrence of series in which the coeffi- 

 cients depend upon the unknown elements. 

 Until the early nineties of the last century 

 the chief aim of astronomers and mathe- 

 maticians had been to modify the methods 

 of Olbers and Gauss by transformations 

 which would increase the degree of accu- 

 racy of the first and the convergence of 

 later approximations. The most success- 

 ful orbit methods would then be those 

 which would yield the elements with the 

 greatest degree of accuracy and with the 

 minimum of numerical work. 



The observations in general furnish three 

 directions of three heliocentric positions of 

 the body, each seen from one of three differ- 

 ent positions of the observer. The problem 

 of the older methods is to pass a plane 

 through the center of the sun which cuts 

 the three dii-ections in such a manner that 

 the body moves in accordance with the law 

 of areas in the conic, which is defined by 

 the three intersections of the plane with 

 the directions, and by the center of the 

 sun. It is evident at once that if the three 

 directions are taken at short intervals they 

 must be given with the utmost precision so 

 that the parameters of the conic may be 

 determined with any degree of accuracy. 



In general a very large number of planes 



satisfying the required conditions may be 

 drawn within the unavoidable errors of 

 observation, so that every preliminary 

 orbit is more or less indeterminate. Thus 

 while a perfect theory might be available 

 for the evaluation of the elements, in prac- 

 tise the numerical accuracy of the orbit wiU 

 be limited. This limitation of accuracy in 

 general increases with the ratio of the 

 errors of observation to observed motion. 

 In addition, even with perfect observations 

 distributed over a sufficiently long helio- 

 centric are, cases occur in which the mathe- 

 matical expressions for the solution of the 

 elements lead to indeterminate forms. In 

 some cases these indeterminate forms are 

 inherent in the physical conditions of the 

 problem. In other cases they may be 

 avoided by proper mathematical devices 

 or by a different mathematical treatment 

 of the problem. One of the best known 

 cases of indeterminateness arising from 

 physical conditions is that in which the 

 orbit plane coincides with the ecliptic. In 

 this case the position of the orbit plane, 

 usually defined by two elements, is given 

 at once, but since each of the three ob- 

 served directions furnishes but one inde- 

 pendent condition, namely the longitude, 

 while four elements remain to be found, 

 the problem becomes indeterminate and re- 

 quires a fourth observation for its solution. 

 One of the best known cases of the other 

 type of indeterminateness arising from the 

 mathematical formulation of the orbit 

 method is the so-called Ausnahmefall (ex- 

 ceptional case) of Oppolzer in Olbers 's 

 parabolic method. "When the orbit is sup- 

 posed to be parabolic only five elements 

 need to be determined from the six condi- 

 tions furnished by observation. The ob- 

 served direction is usually given in right 

 ascension and declination and may be con- 

 sidered as the intersection of two planes 

 which may be introduced as given condi- 



