K. PKARf^ON 15 



Thus e„, e, , e™ aro at oncc foiimi, wlion X„, X, and X, arc known. Probably tlie 

 best way to constnict this parabola is to draw it graphically tliroiigh thc three 

 points 



*'i = -l, 2/1 = 2/0 (e» - ei + e.,), 



a^3 = 0, !/■> = y„e„, 



a^a = ^, 2/3 = 2/0 (e» + e, + e,). 



Gase (iii). Tofit ajmrahola oftlie ihird order to a series of ohservations. 

 Lot thc curve be 



2/ = 2/0 1^0 + e, * + e, (^* j + «3 (^1 



Thc equatioiis to find thc es will now bo 



\. = «0 + 3^2, Xi=^e, + ^63, 



^2 = 3C„ + ^e,, X3 = 5^1 + -feg. 



Hence Co = f (3\„ — 5X.,), e, = ^- {ö\ — TX,), 



e, = J^ (3X, - X„), e, = ^■^- (- 3X, + SX,)- 



Gase (iv). Tofit a parabola of the/ourth order to a series of ohservations. 

 Let thc curvc bc 



Thcn X„= e„ + ^e, + ^64. \ = iei + ic3, 



^ = 5ßo + ie.> + fe4, X3 = iei + }e3, 



Hence we find 



e„ = ^ä (1 5x„ - 70X, + 03X4), e, = -L'i (5X, - 7X3), 

 «2 = W (- ö^^o + 42X, - 45X4), e, = ^{- 3X1 + 5X3), 

 ei = ^ (3X„ - 30X, + 35X4). 



Gase (v). To fit a parabola ofthefifth order to a series of ohservations. 

 Let the curvc be 



y = y„ je„ + e. ^ + e._ i^J + e. (^ + e. g)^ + e. (f )] . 



Tben X„ = e,+ ^e^ + ^e^, X, = Je, + ig, + fe^, 



^2 = 5e„ + ie, + |e4, X3 = ie, + 1^3 + ^e.,, 

 5^4 = ie„ + }e, + 1^4, X5 = f e, + igj + yV^s- 



