64 A New Method of Treating Correlated Averages. 



A solution of this system is afforded by putting a, b, c each 

 proportional to one of the principal minors, and /, g, h each 

 to a certain one of the other three minors, of the following 

 determinant : — 



h Pn> P3i- 



Pl2, 1 />23> 



H P31> P25, 1? 



For call the three minors formed by omitting successively 

 each constituent in the first row a, h, g. By a well-known 

 theorem *, if each of these minors is multiplied by the corre- 

 sponding constituent in the rows other than the first, the sum 

 of these products = 0. 



That is /? 12 a + h -f p 23 g = ") 



p 31 a + p 23 h + g = 3 



Thus the first two of th& above-written six equations are 

 satisfied. By parity of reasoning, if we put, for b, the princi- 

 pal minor b, for c, c, and, for /, the minor f, obtained by 

 omitting the row and column containing (either) p 23 , the 

 remaining four equations are satisfied. The proposed system 

 of proportional values is therefore a solution. And since the 

 equations are simple, it is the solution. 



We have thus obtained by a simpler method than before 

 the solution of the problem f: given the values of some of 

 the variables, what are the most probable values of the other 

 variables ? For the proportionate values of the coefficients 

 «, ?>,... </, h having been ascertained, we have only to sub- 

 stitute in R the given values of one or more variables, e. g. 

 z ! ; and for the most probable values of the remaining vari- 

 ables we have the equations 



\ dx/ z=z > ~ ' \ dy) z=z < ~ 



But we have not obtained a solution of the second problem :J : 

 given the values of some of the variables, what is the law of 

 dispersion for the remaining ones ? For in order to solve 

 this problem we require the absolute values of the coefficients. 

 I do not see how to obtain these, except by the method before 

 adopted, viz., obtaining the integrals of the expression 



J e — (az2+by2+cz2+2f!/z+2crj:z+2hxy) 



with respect to all but two, and all but one, of the variables. 

 Balliol College, Oxford. 



* Salmon, ' Higher Algebra,' lesson iv. article 27. 



f Generalizing the statement given ia the former paper, p. 190. 



X Generalizing the statement given in the former- paper, p. 190 . 



