[ 658 ] 



LXIX. On a General theory of the Method of False Position. 

 By Karl Pearson, F.R.S.. University College. London*. 



(1) XT is in many cases impossible, in others extremely 

 jl laborious, to fit a curve or formula to observations 

 by the method of least squares. I have shown in another 

 placef that the method of moments provides fits which are 

 sensibly as good as those given by the method of least squares. 

 But while the latter method fails to provide a solution in the 

 great bulk of cases, and while the former is much more 

 frequently successful, there still remains a class of cases in 

 which the unknown constants are involved in the curve or 

 function in such a complex manner that neither method pro- 

 vides the required solution. In such cases the following 

 generalization of the " method of false position " will be found 

 serviceable. Apart from practical value, however, the method 

 is of considerable interest as showing a quite unexpected 

 relationship between trial-and- error methods of fitting and 

 the general theory of multiple correlation. 



(2) Let there be a series of observed values Y', Y", Y fn . . ., 

 corresponding to values of another variable X', X", X"'...., 

 and suppose we desire to determine the n constants a, 0,y...v 

 so that 



Y = 4>(X,a,/3,y...v) (i.) 



shall be a curve or formula closely representing the observed 

 facts. 



(Suppose (n + 1) reasonably close trial solutions to be made, 



i. e. (n + 1) false positions given to the curve, and let the cor- 

 responding constants be 



/3. 



«2j ft, 



y, . . . v 



7n • • • "j 

 7 2 , . . . V, 



Un, 0n, V>i, • • V n . 



Let the corresponding values of y, calculated from these 

 trial solutions, be : 



y\ 

 y{, 



y", y'" 

 yi", ///" 

 9,", vi". 



!/»', 



a " ii '" 



and let there be in such 



values used, 



* Communicated by the Author. 



t ' On the Systematic Fitting' of Curves to Observations and Measure- 

 ments/ Biometrika, vol. i. pp. 265-304, and vol. ii. pp. 1-23, 



