8 Mr. G. H, Darwin on Fallible Measures 



wliere the summation is made for the values of a'= — 2, — 1, 

 0, 1, 2, and the correspondino- values of y. In other words, 

 the values of a, h, c, d are to be determined by the method of 

 least squares, so that this curve shall give the best representa- 

 tion of the five points. 



The equations for finding a, b, c, d are therefore 



Da ■j-bl.x + c2).y^ + <i2A'^ = %, 



aXx +b2.v' + c2.v^ + d2,v^ = l^xi/, 



a%v'' + b%^i^ 4- clx^ + dix' = Xxhj, 



a2.v' + bl.x^ + ctx'^ + dtx' = l^xhj. 



From the manner in w^hich the origin has been chosen the 

 sums of the odd powers of x are all zero, and X.r^ = 10, 

 ^x^ = U,^x^ = l?>0. 



Thus the first and third equations ai-e 



5a + 10c= y" +y'-\-y + ^i+.y2, 

 lOa + Uc = Ay'' +y' +y^ + 4.y,', 



and the second and fourth may be easily written down. It 

 will be noticed that the first and third equations would be ex- 

 actly the same if we assumed as the form of the equation 

 y = a'\-bx + cx^. 



Now the proposed method is to substitute for every point of 

 the series of given points the intersection wdth the ordinate 

 of that point of the curve of the form y = a + bx + cx^ or 

 y=a-\-bx + cx^-\-dx^ which best represents that point and the 

 two preceding and two succeeding points. In the case we 

 have been considering w^e are, therefore, to substitute for the 

 the point 0, y the intersection of this curve wdth the axis of y ; 

 that is to say, we are to substitute the point 0, a, because 



when .^' = 



= 0, 



y = a. 











Now 





35« = 



:-3/' + 



12/ + 17y + 12y, 



-%2; 



Ul 





a = 



=2/+A1 



-y'+4y- 



-6y + 



4yi-»/2} 



Thus the correction 8y, to be applied to y, is — ^ A^y^^ 



Hence generally, since the process is supposed to be applied 

 all along the series, 



To give a geometrical meaning to the rule, it may be observed 

 that 



and therefore if A' be a symbol denoting the operation of dif- 



