METHODS OF INTERPOLATION. • 291 



and (16) may be written — 



10 Z-«4 = />-[4( Ih-j-'U, + U,) — («2+ ?^6)] 



Adding these two equations, we obtain — 



Since J: may have any vahie, let us determine it so that the excess of 

 the weight of u^ J^ud n^ over that of «2 i^ud Hg shall be equal to the ex- 

 cess of the latter weight over that of Ui and j/^. This gives — 



l-j_|_4A:_(S_A-) = S — 7>--f5 



and, consequently, A- ==3. The formula then becomes — 



iu=sh\^H>h+i'4+>(,)+Hih+K6)—'Xu,+i,,)] . . . (IS) 



and, if the weight of the middle term is increased by 7, we Imve 

 tiiially— 



"4 = 4t[1'^"4 + 11(»3+''5) + 4(''2+"c) — 3(«l+?M] . . . (10) 



Here the weights increase in arithmetical progression, from the extreme 

 terms to the middle one. 



To obtain a similar formula including nine terms, we may proceed as 

 follows. In a series of the third or any lower order the fourth differ- 

 ences are zero, and any five consecutive terms are connected by the 

 relation — 



In a series of the fifth or any lower order the sixth differences are zero, 

 and for any seven consecutive terms we have the relation — 

 i<7 — Uq -f 15 U5 — L'O W4 + 15 »3 — «2 + «i = 



In a series of the seventh or any lower order the eighth differences are 

 zero, and any nine consecutive terms are connected by the relation — 



W9 — S ?/8+2S ?/-— 50 »G + 70 «5 — 5G Ui+2S i^—S «.+ «i = 

 Hence, considering any nine consecutive terms in a series of the third 

 or any lower order, we hdve — 



35A-t<5 = 15/.-(?^4 + «5 + «G) — 0/.-(?f3 + ^'7) + /''("2 + ^M 



10 /c'm5 = 4 /.•'( 7^4 + «:,+ ;/,;) — /'•' (":i+ "7) 

 Adding these three equations together, we obtain — 



(lL>0+35A--flO//)»5 = (5G + 15A:+4/y)(W4 + «5+"G) — (2S+r,A-+A-')(W3+?M 



+ (8 + A-)(M2+W8) — (Ml + «9) 



which expresses a general relation between an}- nine consecutive terms 

 in a series of the third or any lower order. The numbers A* and Jc' being 

 entirely arbitrary, we may make the coefficients in the second member 

 of the equation form an arithmetical progression by takiug — 



(8+A-)-f2(2S+GA-+A') + (50+15A-+4A-')=0 

 _ 1 _ 2(8 -f A-) — (28 + G A-+ A') = 



