Adjustment of Observations. 223 



problem us given above, it may be interesting in conclusion 

 to apply it to one or two specific cases. 



The first is an example taken from Memmaii*, and has been 

 chosen partly as illustrating the occasional misapplication o£ 

 the method of least squares mentioned earlier in this article. 

 In introducing the example Merriman says : — " In the last 

 article the quantities y and x were both observed ; but the latter 

 was regarded free from error, because .... The following 

 example gives an outline of a method that may be used when 

 both observed quantities are affected by accidental errors. — 

 In order to determine the most probable equation of a certain 

 straight line, the abscissae and ordinates of four of its points 

 were measured with equal precision. The observed co- 

 ordinates are 



y = 0-5, 0-8, 1-0, and 1% 



x = 0-4, 0-6, 0-8, and 0-9. 



And the most probable straight line for the four points 

 is expressed by the equation 



y = S# + T 



in which S and T are constants whose most probable values 

 are to be found. " 



Then, notwithstanding his preamble, he proceeds to a 

 solution which is the exact equivalent of the assumption 

 that the measured values o£ x are free from error. That is, 

 he forms normal equations which are the equivalent of 



txy= 82tf 2 + T2^ 



ty = SZx + 4T, 



obtaining the solution 



S == 1-339, 



T = --029. 



Had the equation of the line been written in the equivalent 

 form 



I T 



and a similar solution made, the results would have been 

 S = 1-354, 

 T = --039. 



* ' Theory of Least Squares' (original edition), Art. 107. 



