fxvii, 



Theory of the Method of False Position. 663 



We have by (xiv.) : 



— Rot'(r = -Tydt'a<T t 'R 00 + -fy-dt'po-p'R 00 + • • ■ + ~fY~ dfv<rt'll 00 . (xv 



since t—n 



S (ptt'i'st)=0 unless s = t f , and then it =R 00 . 



Rearranging we have : 



Rot cr \a &J3 A o v ... 



~ r^^ = iy df ' a + d *'^ + • • • + tt^' 1 " • (xv, - j 



Multiply by A^, A 2 e, . . . A„e the equations obtained by 

 writing £' = 1 to /? respectively. We find finally : 



This is the required result, and appears to be a very remark- 

 able one. 



(3) W T e notice that : 



(i.) The quantities in round brackets are the well-known 

 partial regression-coefficients of the theory of multiple cor- 

 relation. 



(ii.) The form of the function used is not directly involved 

 in (xvii.), the coefficients being solely functions of the observed 

 and trial solutions. 



Hence, if the trial curves be given by the use of a me- 

 chanism which involves s degrees of freedom in its placing 

 and setting screws &c, s + 1 trials will give us by the method 

 of false position the best position and setting of the mechanism 

 to strike the closest curve. In this case the actual mathe- 

 matical form of the function may be unknown or unknowable *. 



(iii.) The multipliers of the constant-differences Aje, A 2 e, &c. 

 are absolutely the same, whatever constant we are seeking. 

 Hence, if they are once determined numerically, however 

 many constants there are in the formula, no additional trouble 

 is involved. For example, in fitting a circle to n arbitrary 

 points, the correction of its radius on the reference -circle 



* For example, it is a common practice with draughtsmen to till in a 

 curve through a series of plotted points by aid of a spline bent through a 

 series of arbitrary points obtained by the sharp vertical edges of weights 

 placed on the drawing-board. Each such edge has two degrees of free- 

 dom. Hence given m such weights and the spline, 2m+l trial solutions 

 would by the method of false position give the position for the weights 

 to get the best spline curve through the observations. Of course such a 

 process would be using a steam-hammer to crack nuts, but it will suffice 

 to suggest how perfectly our result is freed of mathematical function or 

 hvpothesis. 



2 Y2 



