330 METHODS OF INTERPOLATION. 



iii^ Ml terms each, and equidistant, so that h may denote the constant 

 interval between their middle points, and if we put A'=Awi, and — 



and phice the origin of coordinates at the middle of the left-hand group, 

 then the sums of the terms in the several groups will be — 



So=A'+B'+C'+D'+&c. 



Si=A'+B'/5*+CV''+B'«^"+&c. 



S2=A'+B';S2*+Cy"+D'r52''+&c. 



&c. &c. 



and in any given case, assuming as many groups as there are constants 

 to be determined, we can find the values of the constants from these 

 equations of condition, just as in ordinary interpolation from ordinates. 

 In accordance with the general method referred to, we proceed as fol- 

 lows : If the number of constants is an even one, for instance, six, the 

 groups forming a recurring series of the third order, whose scale of rela- 

 tion is — Ao, — Ai, — Ao, we shall have the three equations — 



A0S0+ AiSi-f A2S2+ 83=0 

 AoSi+AiS2-fA2S3+S4=0 



AoS2+AiS3+A2S4+S5 = 



These enable us to find the numerical values of Aq, Aj, A2, and we substi- 

 tute them in the equation of relation — 



z^+A2Z^+AiZ+Ao=0 



This numerical equation of the third degree being solved, its turee roots 

 will be the values of the three constants /S*, y'', 5\ Substituting them 

 in the three equations of condition — 



So=B'+C'+D' 

 Si=B'/S*+CV4-D'<5'^ 



S2=B'/s2*+cy"+D'(52* 



we can find the values of B', C, and D', and consequently those of B, 0, 

 and D. Having thus determined all the constants in the equation — 



we are enabled to interpolate the sum S of any 7i terms taken in a group, 

 or any single term, and to form a recurring series of the third order, 

 such that the arithmetical means of the terms in the six assumed groups 

 will be the same in it as in the given series. The equation of the grad- 

 uated series will be of the form — 



n='B"i3''+C'Y+'D"d' 



When the assumed groups are consecutive, w© shall have 7i=7h. The 



