Theory of the Method of False Position. GG7 



The coefficients in r of the regression equation are now 

 respectively : 



r 01 ( 1 - ?| 3 ) - r 02 (r 12 - r sl r 2Z ) — r 03 (r 3l — r 2 r 23 ) 



L— 



'>?> 



i 2 + ~2r 2Z r 12 r H 



= •072,327, 



02 



( 1_ r ll) —^03(^23 — ^l2^3l) —^0l(^12-^23^3l) 



1-^3 



^03(1 — r? 2 ) " r 0l(^81 - ? '23>12) —^02(^23-^3^12) 



1 - r 23— ? 1l~ r ?2+ 2r 23?*12? 



•692,501, 



= •318,083. 



Whence the general numerical formula for modifying the 

 constants of the reference-circle is : 



A u e = -093,522A 1 e + -602 5 783A 3 e + -255,849A 3 e. 



Putting e~h, k, r successively we have for the circle of 

 approximately closest fit : 



/* = 2'562, ife =— -689, r =.2"657. 



The accompanying tahle gives the ordinates, or differences 

 found from this approximate circle of closest fit, and from the 

 four trial circles. 



x=0 ... 

 x=l ... 

 x = 2 ... 

 x=3 ... 

 x=± ... 



Ordinates. 



Differences. 



Ob- 

 served. 



o-o 



1-5 



1-8 

 20 

 1-5 



Closest 

 Circle. 



•015 

 1-460 

 1-908 

 1-932 

 1-545 



Closest 

 Circle. 



1st 

 Trial. 



2nd 

 Trial. 



3rd 

 Trial. 



4th 

 Trial. 



+ '015 

 -•040 



, +-108 



1 --068 



+ •045 



■000 



+ •344 

 + •391 

 + •049 

 -•235 



+ •386 

 -l-217 

 +■281 

 -•004 

 -•106 



-•058 

 -•075 

 + •111 

 -•016 



+•180 



•000 



-129 

 -032 

 -•273 



-•279 



Square Root of Mean 1 

 Square Difference J 



•063 



•256 



•239 



•104 1 -184 



It will be observed that tested by the square root of mean 

 square of the differences, the circle obtained by this method 

 of false position is 3 to 4 times as good as all but one of the 

 trial circles. The third trial was a peculiarly lucky one, but 

 even here the false-position circle is more than half as good 

 again. The accompanying diagram gives the four trial 

 circles and the false-position circle. If we wanted a still 

 closer approximation, we should now throw out the worst of 

 the trial solutions, i. e. the first, and work from the first 



