March 17, 1905.] 



SCIENCE. 



421 



Lara's conclusions and the writer's; and 

 also to show how one of the former's most 

 interesting results can be obtained in a 

 manner different from that used by him. 



The writer's theorem is: "Let there be 

 given two series of observation equations 

 as follows: 



aix + b^y + c,2 + ... ^ «! — 0, I , 

 -t- b.,y + c,z + ... + 7i.^ = 0, j 



ctix + 6i2/-i- + ••• + + ••• + "i = Oi j 

 OjZ + b^y + c^z + ... + PjU) + ... + Hi = 0, J 



the equations being identical in the two 

 series except for the addition of one or 

 more new unknowns w, ••■ in (2). Let 

 each of these series of equations be solved 

 by the method of least squares, and let: 

 \vv\-i, be the sum of the squares of the 

 residuals resulting from the solution of 

 equations (1); [vv].,, be the sum of the 

 squares of the residuals resulting from the 

 solution of equations (2) ; then, no matter 

 what may be the law of the coefficients 

 Pi) P2> ' ' "j fiiitl even if these coefficients are 

 assigned at random, [vv]^^, is always 

 larger than [t'l'Ja-" 



The conclusion drawn by the writer from 

 this theorem is as follows : 



"The method of least squares is used 

 ordinarily to adjust series of observation 

 equations so as to obtain the most probable 

 values of the unknowns. But there is a 

 subtler, and perhaps more important use 

 of the method; when it is employed to 

 decide which of two hypothetical theories 

 has the greater probability of really being 

 a law of nature ; or to decide between two 

 methods of reducing observations. In 

 such cases, astronomers not infrequently 

 give preference to the solution which brings 

 out the smallest value of [vv] , the sum of 

 the squared residuals. But in the light of 

 the above theorem, it becomes clear that 

 the mere diminution of [vv] alone is in- 

 sufficient to decide between two solutions. 



when one involves more unknowns than 

 the other. To give preference to the sec- 

 cud solution, it is necessary that the dim- 

 inution of [vv] be quite large, and that 

 the additional unknowns possess a decided 

 a priori probability of having a real ex- 

 istence." 



In his paper in Astr. Jour., 521, Mr. 

 Midzuhara says: "This conclusion, per- 

 haps, depends on the author's misappre- 

 hension of the principle of probability. 

 For I believe that to compare the prob- 

 abilities of the two solutions we must neces- 

 sarily take 



-~ — - and 



where m expresses the number of observa- 

 tions, and jjL^ and fi.^ are the numbers of 

 the unknown quantities in the first and 

 second solutions, respectively." 



In other words, Mr. Midzuhara takes as 

 the criterion for deciding between the two 

 solutions the quantity ordinarily called 

 'mean error of one equation,' instead of 

 the sum of the squared residuals. When 

 the number of unknowns in the two solu- 

 tions is different, these two criteria may 

 give opposite results ; the one indicating the 

 first solutions as the more probable, the 

 other, the second solution. 



It is evident that practise of astronomers 

 varies in this matter. Mr. Midzuhara, for 

 instance, and doubtless other astronomers, 

 too, use [vv]/in — as the criterion. On 

 the other hand, Bessel was in the habit of 

 using [ vv] . A good example is to be found 

 in his classic paper on the parallax of 61 

 Cygni. He there* reduces his observations 

 with parallax terms, and again without 

 them. He decides in favor of the reality 

 of his parallax terms solely on account of 

 the diminution of [vv] ; and not until after 

 this is decided does he compute the mean 

 error \/[vv]/m — ii. This quantity he 

 calculates for the parallax solution only, 



* Astr. ^'ach., No. 366, p. 87. 



