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SCIENCE 



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



we have a significant contrast of the inter- 

 pretations with reference to accuracy ob- 

 tainable from theory and practise. Theo- 

 retically small perpendiculars in Olbers's 

 method would lead to indeterminateness, 

 but in practise it is not the absolute magni- 

 tude of the small perpendiculars that 

 counts so much as their uncertainty due to 

 errors of observation. To return once more 

 to Olbers's method, when his mathematical 

 formulation leads to a practical indeter- 

 minateness the difficulty may be removed 

 at once by substituting for Olbers's great 

 circle through the second place and the 

 sun a great circle perpendicular to it. This 

 choice of great circle evidently produces a 

 maximum value of the perpendiculars 

 drawn to it from the first and third places, 

 so that the effect of the errors of observa- 

 tion is minimized. 



It must not be supposed that the condi- 

 tions of indeterminateness just referred to 

 were not known to theoretical astronomers 

 until recent times. In his classic "Lehr- 

 buch zur Bahnbestimmung, " the second 

 edition of which was published in 1882, Op- 

 polzer sets forth clearly and concisely the 

 significance of errors of observation with 

 reference to small quantities which are 

 theoretically of a high order of smallness, 

 when the intervals or motion are considered 

 quantities of the first order. My own aim 

 and that of those associated with me at the 

 University of California has been to treat 

 each case on its own merits from the nu- 

 merical point of view and to ascertain at 

 the outset the uncertainty which must nec- 

 essarily remain in the result. As this un- 

 certainty corresponds to a region of pos- 

 sible numerical results clustering around 

 the physical solution or in case of multiple 

 mathematical solutions around these, all of 

 which correspond to orbits that will satisfy 

 the observations within their errors, I in- 

 troduced the term 7-ange of practical solu- 



tions in a paper read at the International 

 Congress of Arts and Sciences at St. Louis 

 in September, 1904, and have at the same 

 time and again later set forth the numer- 

 ical conditions producing a range of prac- 

 tical solutions. In the modifications of the 

 formulffi, for computing orbits so as to se- 

 cure the greatest accuracy with the least 

 expenditure of numerical work this prin- 

 ciple has been constantly borne in mind. 

 I emphasize this point because this distinc- 

 tion between practise and theory 'has not 

 been well understood. Moulton, in a very 

 eslaaustive memoir on the "Theory of De- 

 termining Orbits," published in the Astro- 

 nomical Journal in 1914, to which further 

 reference will be made later, also seems to 

 have failed to recognize the significance of 

 our work in this regard, although it was 

 set forth in detail in another form in 

 Buchholz's "Klinkerfues Theoretische As- 

 tronomic," third edition, 1912, which 

 jMoulton has reviewed. In the first example 

 published in this work I was careful to 

 place a dot over the last digit of every 

 fundamental quantity that could be relied 

 upon. 



To facilitate our further discussion it 

 may be well to trace in outline the funda- 

 mental principles of the methods of Olbers 

 and Gauss as set forth by Oppolzer in a 

 masterly manner. Olbers's and Oppolzer 's 

 parabolic methods yield a solution of the 

 first and third' geocentric distances from 

 the equation fm = MPj -\- m and the weU- 

 known Euler's equation expressing the 

 intervals between the first and third dates 

 in terms of the sums of the radii vectores 

 drawn from the sun to the first and third 

 places and the chord joining these two 

 places. With Olbers's choice of the great 

 circle through the middle place m becomes 

 zero. In both methods the ratios of the 

 triangles are replaced in the first approxi- 

 mation by the ratios of the intervals. Even 



