8 On the Systeinatic Fitting of Ctirves 



Wc shall limit ovir discussion to equal base elemeuts b, whioh caii ahvays be 

 takeu as uuity. 



Let v,:=S{x"y). 



Then the following expression for thc »th moment A/„' of thc trapezoidal area 

 about the axis of y is easily deduced : 



,., ,. «.(n-1) , n(»-l)(n-2)(«-3) ., 



«(»-l)(»-2)(n-3)(»-4)(»-.5) 

 + — 2ÖT6Ö '^"-'^ "^••• 



■^"1(2'' |3 " ^ [4 " ■ 



Hei-e the " corrective terms" in ^„, and y„ are nothiug niore thau thc 

 subtraction of Ihc ni\\ monients of the triangles PQR and STV from the »th 

 moiiicnt of the whule figure oii base RS, which is represented by the remainder 

 of thc expression. They can bc thrown iiito the simple forms 



(a;„ + 6)"+' - {n + 2) a;^"+' h - x-J'^^ 

 ^"' (n + !)(« + 2) 



_ (- a-, + 6)»^' - (n + 2) (- ^o)»+' fe - (- x,)"*' 



^ ' ^° (?^+l)(n + 2) 



Now let PT= 21 and let us takc moments about the middle of PT. Let Jtf„ 

 be the nth moment abont this point, and x being measured from it, let 



m 



Vn = S{x''y). 







We havc x,n = —x„ = l, 



and accordingly, if / bc measured in b as imit, 



« (h -1) «(«-!)(»- 2 )()t - :$) 



M„ = I'„ + -^ 1 „ — I'n-a H US?; "«-4 



w(w-l)(»-2)(w-3)(w-4)(«-5) 

 + 20160 ">.-« + 



(n + l)(« + 2) t^^^^^ ^^ ^'''• 



The corrective term is thus very simple : il may be written 



•in(i/m+(-l)"»/o). 



