664 Prof. Karl Pearson on a General 



radius will be given by exactly the .same formula as the cor- 

 rections for the coordinates of its centre. 



The various uses of the formula (xvii.) can only be briefly 

 indicated here. 



It arose from the consideration of a special physical 

 problem. A somewhat complex formula for astronomical 

 refraction had been obtained which involved for given 

 meteorological conditions one arbitrary constant only. How 

 was the value of this to be determined from the observed 

 values of refraction at different altitudes ? The direct 

 application of the method of least squares was idle ; the 

 constant was involved in far too transcendental a manner for 

 such a method to be of service. Accordingly two trial solu- 

 tions were made, and the values of cr o , o^, and r 0l found; then 

 the correction of the constant, e — e, is given by 



^o-e = >ui J( e i- e ) .... (xviii.) 



where e and e 1 are the two trial values, and e is taken as the 

 reference trial. 



This corrected solution has again to be taken as a trial 

 solution with the better of the two trials, and thus a very close 

 value for the constant in question can be determined. 



Clearly the only calculation involved in (xviii.) is by (ii.) 

 and (iii.) : 



. a <> s (ffo-y)Cyi-y) 

 01 n' ®(yi-yr 2 ' 



Formula (xviii.) immediately led to its generalization for 

 two unknown constants determined by three trials, i. e. 



6 o ~ 6 = 1 -* — IT Ol ~ 6 ) + 1_> - ( 6 2 - • (™.) 

 17 12 °1 L 7 12 ff 2 



and this ultimately to the complete generalization given in 

 (xvii.). 



If n+1 trials are used to determine one constant, then it is 

 easv to see that the best result will be obtained by using; 

 (xvii.) straight off*. 



Another service which, 1 think, can be performed by the 

 method of false position is of the following kind. It is well 

 known that the accuracy of both physical and astronomical 

 investigations can be largely influenced by temperature, 

 pressure, or hygrometrical conditions. What are the most 

 suitable conditions to cany on a particular class of observa- 

 tion under ? Let such conditions be represented by a, /3, 7. 

 Then make four trial sets of observations of the kind under 



