580 



SCIENCE 



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



If, therefore, for the present, we assume 

 the coordinates a, 8, their velocities a', 8', 

 and their accelerations a", 8" to be known, 

 Ave have three fundamental equations for 

 the solution of the four unknowns p, p, p", 

 and r. The fourth equation is derived 

 from the triangle sun-earth-object, and in- 

 volves p, r, and known quantities. By 

 elimination the problem reduces to La- 

 grange's equation of the seventh degree 

 with not more than two positive real roots, 

 which may be interpolated to six decimals 

 from a table which I have prepared for this 

 purpose, so that the solution may be ac- 

 complished without the hitherto necessary 

 laborious numerical approximations. 



The direct solution which has just been 

 outlined corresponds to the so-called first 

 hypothesis of other methods. It is evident 

 that the accuracy of Laplace's direct solu- 

 tion depends upon the accuracy of the 

 fundamental observational data for which 

 we have chosen a, 8, a', 8', a", 8". If the 

 epoch is chosen to coincide with the date of 

 one of the observations, then a, 8 are fixed 

 numbers, and the accuracy of the Laplacean 

 solution depends upon the accuracy of the 

 adopted values of their velocities and accel- 

 erations or, which is an equivalent state- 

 ment, upon the accuracy of their first and 

 second differential coefficients. In prac- 

 tically all other methods the accuracy of 

 the solution depends upon the accuracy of 

 the adopted values of the ratios of the tri- 

 angles. Unfortunately the method of La- 

 place has been prejudiced by Lagrange 

 until recent times through a letter ad- 

 dressed to Laplace, in which he says 

 that while analytically Laplace's method 

 constitutes the simplest solution of the 

 problem, in practise it does not afford 

 corresponding advantages because the dif- 

 ferential coefficients could not be deter- 

 mined with the necessary accuracy. This 

 far-reaching statement Lagrange intended 



as a mere opinion which he proposed 

 to verify later by mathematical demonstra- 

 tion. It is a remarkable fact that La- 

 grange's opinion, although never verified 

 by himself, has been the chief cause of re- 

 tarding the further development of the 

 Laplacean method until recent times. 



Nevertheless, several attempts at giving 

 it a practical formulation have been made 

 during the last century, but with indifferent 

 success. With reference to the disrepute 

 which Laplace's method and all formula- 

 tions based upon the same have been held 

 until recent times and for a statement of 

 its actual merits I may refer you to my ad- 

 dress delivered before the International 

 Congress of Mathematicians in August, 

 1912. Laplace's method leads to the usual 

 equation of the seventh degree, which, as 

 stated above, we shall refer to as Lagrange 's 

 equation. The roots of this equation have 

 been frequently studied by Cauchy, Mrs. 

 Young (Grace Chisholm), Oppolzer, and 

 others. In recent times a classic study of 

 the equation has been published by 

 Charlier, who not only clarifies the exist- 

 ing conditions which will lead to a mul- 

 tiple solution, but exhibits these condi- 

 tions geometrically by dividing space into 

 four regions symmetrical with reference to 

 the line joining the earth and sun as cen- 

 tral axis, and showing that two solutions 

 exist when the body is in two of these 

 regions, and one solution when the body 

 is in the other two. In certain cases it 

 is not possible to distinguish the mathe- 

 matical from the physical solutions, so 

 that either a fourth observation must be 

 employed in the original solution or the 

 mathematical solution must be eliminated 

 on the basis of difference between theory 

 and later observation. 



My own formulations of Laplace 's method 

 need be referred to but briefly. The re- 

 sults are chiefly that the whole process has 



