186 Dr. Norman Campbell on lite 



we shall assume the usual results of L.S., namely, that 



2y 



dx 2 =— , where Xv 2 is the sum of the squares of the 



JN — n x 



d x * 



residuals, and dX 2 = — T . 

 p-1 



Examples. 



It will now be well to show how the proposed method 

 works out in practice. The chief difficulty is in the 

 selection of material. I have a large amount of matter 

 of my own to which it might be applied, but the use of 

 that matter might not inspire confidence. Observations 

 are not usually published in sufficient detail to enable them 

 to be recalculated ; so three examples have been taken 

 (although they are not wholly suitable, because the number 

 of observations are so small) from the 8th edition of 

 Merriman's ' Method of Least Squares/ pp. 126, 132, 138. 

 The advantages of the method in the saving of labour increase 

 rapidly both with N and n, but even in these simple cases 

 they are enormous. In L.S. £Nn(n + 1) multiplications have 

 to be performed, and then ^(« + 2)(n + l) columns, each of 

 N entries, added ; in Z.S. there is no multiplication, and 

 only ~N(n + l)/n columns, each of N/w entries, are added. 

 Using a calculating machine, to which I am thoroughly 

 accustomed, omitting all " checks " (and the omission wastes 

 time on the whole), and reducing writing to a minimum by 

 keeping figures on the board of the machine, I found that 

 the mere writing, quite apart from calculation, involved in 

 the formation of the normal equations of L.S., occupied 

 longer than the complete formation of these equations 

 by Z.S. The solution of the equations takes the same time 

 in either case : it takes longer than the formation of the 

 normal equations by Z.S., but not nearly as long as that 

 by L.S. 



In each example there is given (1) the equation which the 

 observations have to satisfy ; (2) the observations ; (3) the 

 normal equations and solution by L.S. ; (4) the normal 

 equations and solution by Z.S. ; (5) in the last four columns 

 of the observations, the residuals and their squares according 

 to L.S. and Z.S. The observations which are added to give 

 the normal equations of Z.S. are bracketed. In the third 

 example the equation (1) is not linear : for the purposes of 

 calculation it was reduced to linear form, in accordance with 

 the usual practice, by taking logarithms of both sides ; these 

 logarithms, used in the calculation, are given in the table. 



