, 2'+^ {i + ^)"+" 



STORY. — A NEW GExNERAL THEORY OF ERRORS. 193 



(84) = S„„_o - s ■ &„,_! for ^ m, 



, • &. =^_(- 1 ,. ( . ) [2 », + 1 - 2 (™ - 0] jsr(7qr|y5«, 



m— 1 



= (2m + 1) • 5,„.^i -2m^(- ])•/'" ~ 



■' ^ 



= (2 m + 1) • S^rn+l — '^ m • S.2m-l , 



that is, 



9 tMt 1 



(85) S.^„+i = 2,,^^ I ' ^-^'"-i + 2 ,„ + 1 ■ « ' -^2^ for ^ m. 

 From (68), (84), and (85) we obtain 



f R^,,, = R2,i,-2 ~ * ■ R-2m-\ for ^ w, 



(^6) 1 2 m 1 



2 »i + 1 2 W2 + 1 ^ 



while from (G8), (83), (84), and (85) follow 

 ?^ = - (2 m • ,S„„_i + s • ^2,„) e~^ = - (2 m + 1) &„ , i ■ c"'^ 



(87) =_(2m + 1) -i^^^+i, 



ds 



(88) =B„„^,. 



From (83), (87), and (88) follow, after simple redactions by (84)-(86), 



9 



X- {'So„+i i?2* — *S'2m -^2i+l} = 2 (??« — k) • S.2m+1 -^2^+1 , 

 C'S 



^J2m- ^2.-1 ^2.+i + S,„ R,,} = (2m-2k-l)- .%„, R,,^, . 



Therefore, letting [ ] denote the excess of the value of the enclosed 

 function of s for s =^ s over its value for s = s, we have 

 J 



(89) fs,.,^ i?3. • ^^ = ^ [S,^-^ RJ^ - ^^ ^^- ^--]! 



VOL. XL. — 13 



