June 8, 1917] 



SCIENCE 



579 



comet orbits would not have been set down 

 as being parabolic and much anal}i;ical 

 work with reference to the origin of comets 

 would have been avoided. Yet there is a 

 practical advantage in adopting a prelim- 

 inary parabolic solution for a comet when 

 the range of solutions is very large and 

 when this range includes the parabola. 

 For since the majority of comet orbits have 

 high eccentricities the adoption of a pre- 

 liminary short-period orbit would later in- 

 volve a more radical correction than the 

 adoption of a preliminary parabola. Of 

 course within the physical indeterminate- 

 ness or the practical range of solutions any 

 and all of the orbits satisfying the observa- 

 tions within their errors are equally justi- 

 fied, but even to this day it is a reflection on 

 the astronomer if the period or eccentricity 

 of his preliminary orbit must be increased 

 to satisfy later observations, while it is 

 quite the proper thing to publish a para- 

 bola and later to find the orbit short pe- 

 riod even if such short period and eccen- 

 tricity could have been derived at the out- 

 set. Thus in the case of comet Neujmin 

 referred to above parabolic orbits were still 

 insisted upon, while the comet closely fol- 

 lowed our short-period orbit from a two- 

 day arc. It has been my frequent experi- 

 ence that elliptic orbits with a fair degree 

 of accuracy could be determined from the 

 first three observations, while other com- 

 puters continued to produce parabolas 

 which could be shown not to lie within the 

 range of solution and resulted from the 

 use of approximate methods. 



In anticipation of stating the many ad- 

 vantages introduced by the modernization 

 of the method originally proposed by La- 

 place I have already dwelt on three im- 

 portant considerations, namely, first, on the 

 complications involved in the successive 

 hypotheses and approximations of the older 

 methods; secondly, on the waste of time 



in applying different formulas for a condi- 

 tioned and a general solution so that with 

 the abandonment of a parabola it is neces- 

 sary to make a new start; and thirdly, on 

 the significance of the range of solution as 

 derived from partial indeterminateness de- 

 pending upon the shortness of the arc and 

 upon the effect of error of observation on 

 small significant coefficients. My own at- 

 tention was directed to Laplace 's method by 

 a memoir of Harzer. Laplace's method ap- 

 peared in 1780 in a memoir and later in his 

 Mecanique Celeste. Prior to him, in 1771 

 Lambert had produced his famous theorem 

 based on geometrical considerations. Later, 

 in 1778 Lagrange showed that Lambert's 

 equation leads to an equation of the seventh 

 degree, the fundamental equation of the 

 orbit problem which also occurs in the older 

 methods and which Charlier proposes to 

 designate as Lagrange's equation. Laplace 

 and Lagrange mutually inspired each other 

 to further important developments of the 

 orbit problem. Laplace starts with the 

 three differential equations of motion of the 

 second order for the two-body problem 

 under the Newtonian law of attraction as 

 applied to the motion of a material point 

 (the object) about the sun. He then ex- 

 presses the heliocentric rectangular co- 

 ordinates of the object in terms of its geo- 

 centric coordinates and the heliocentric 

 coordinates of the earth. The rectangular 

 coordinates are then replaced in terms of 

 polar coordinates, and thereby three equa- 

 tions are derived which give the geocentric 

 distance P, its velocity p', and its accelera- 

 tion p" at the epoch in terms of the ob- 

 served coordinates (for which we may 

 choose a and 8, their velocities (a' 8'), 

 and their acceleration (a", 8"), and also 

 in terms of the unknown heliocentric dis- 

 tance r of the object and known quantities, 

 depending upon the motion of the earth 

 about the sun. 



