484 Mr. G. U. Yule. On the Significance of Bravais Formula? 

 Our characteristic or regression-equation will now be of tlie form 



6 13 and 6 13 being the unknowns to be determined from the observations 

 by the method of least squares. I have omitted a constant term on 

 the right-hand side, since its least-square value would be zero as 

 before. The two normal equations are now 



or replacing the sums by the symbols defined above, and simplify- 

 ing 



= 6 12 <r 2 -f &i 3 r 33 03 1 



whence 



-* - 2 



(11). 

 &13 = " 



That is, the characteristic relation between %i and x*x-s is 



Now Bravais showed that if the correlation were normal, and we 

 selected a group or array of Xi's with regard to special values h z and 

 h 3 of #2 and # 3 , then 7^ being the deviation of the mean of the selected 

 Xi's from the X r mean of the whole material, 



where & 12 and 6 13 have the values given in (11). But evidently the 

 relation is of much greater generality ; it holds good so long as ^ is 

 a linear function of 7i 2 and & 3 , whatever be the law of frequency. 



Further, the values of b iz and & 13 above determined, are, under any 

 circumstances, such that 



is a minimum. If we insert in this expression the values of 6 12 and 

 6 13 from (11), we have, after some reduction, 



(13), 



