LAPLACE. 581 



x' = Xt [qi + h) aji + ^lt (qi + k,) hij\ + vt {qt + h)CiJi + . . . 

 = 2 (^. + h^ji [Xtti -{■ lxhi + vCi+ ...] 

 =^tyi{qi + k) by (7). 



The advantage of using equations (10) is twofold ; in the 

 first place we determine x, and thence x, by a systematic process, 

 and in the next place we see that the equations (10) are sym- 

 metrical with respect to x, y , z, ... : thus if we had proposed 

 to find y, or z, or any of the other unknown quantities instead of 

 X, we should, by proceeding in the same manner as we have 

 already, arrive at the same system (10). Hence the same ad- 

 vantage which we have shewn by the Theory of Probability to 

 belong to the value of x by taking it equal to x, will belong 

 to the value of y by taking it equal to y, and to the value of z 

 by taking it equal to z, and so on. In fact it is obvious 

 that if we had begun by investigating the value of y instead of 

 the value of x the conditions (6) would have been changed in such 

 a manner as to leave the proportion of the factors y^, %, y^^, "• 

 unchanged ; and thus we might have anticipated that a sym- 

 tnetrical system of equations like (10) could be formed. 



We have thus shewn how to obtain the most advantageous 

 values for the required quantities x, y, z, ... 



Suppose now that we wished to find the values of x , y , z\ ... 

 which render the following expression a minimum, 



'%f [ciiX + hiy + CiZ -i- ... — $'i - ^if; 



it will be found that we arrive at the equations (10) for deter- 

 mining X , y , z ... Hence the values which have been found for 

 X, y, z, ... give a minimum value to the following expression 



^ji {€i — kiY that is Xf ^^-T—j ' 



If h^ be zero, and hi constant, for all values of t, the values which 

 have been found for x, y, z,... render the sum of the squares of 

 the errors a minimum : as in Art. 1007 these conditions will hold 

 if the function of the facility of error is the same at every ob- 

 servation, and positive and negative errors are equally likely. 



