General Theory of the Method of False Position. Qt)S) 

 For brevity write : 



^=^{yv-yf (»•) 



where 



Sfe-i/) 2 = (y,.'-y') 2 + (y,,"-y"y + (yv'"-y"7+ , ■ ■ 



and 



<r v Xa pi XT pp ,= --$(y v -y)(y v ,--y) . . . (iii.) 

 in 



where 



S(yr-y)(y P '-y) = (y/-y%y/-yl + M'-y'')(y/~.>n 



+ W'-y'")(nj"-y'") + - 



Let the actually observed values be, yj yj', y Q rfl '..., and the 

 best values of the constants for these values : 



««, Po, yo,-v Q - 

 Then, clearly, if p and p' take any values from 1 to ?i, 

 it is merely straightforward arithmetic to discover the 

 numerical values of any a P and r ,. Let 



Y = <£(X, « , O , 7 , ...v ) 

 be the required formula. Then, by the method of least 

 squares ; we require to make 



a minimum by varying- a 0) fi , y , y .v . 



Now by " reasonably close trial " solutions, I intend to 

 convey that any series of constants <x Vy ft p , y P ,...v p differ by 

 fairly small quantities from the " best values." Hence we 

 shall consider the differences a p -*a , /3 P — /3 , y p — 7 ,...v P — v 

 so small, that to a first approximation their squares may be 

 neglected. The whole process may, however, be repeated 

 when a very close degree of approximation is required, by 

 taking a series of fits with small divergences from the first 

 approximation. We have to our degree of approximation 



Y=0(X, « + «„-«, £ + &-£. y + y,-y,... v + v -v) 



where 



A p a = a P — a, &C. 

 Further let us write 



d(f) d(j) 



d<t>jcU = Ca, di3 = C fr •'• dv = <>. 



Then we have to make 



U = S(y-y 4 c«A « + c^AjS + . . . + c v A v) 2 . 

 IS denoting a summation of y, //,, through all the possible 



