STORY. — A NEW GENERAL THEORY OP ERRORS. 



183 



If we write 



f 



(48) 



P;„. = 



P.,. 



(- 1)' 



Pa 



/;2'(*-i)<' 



(•■) 



and 



2a- 



0' 



P2i+l 



we have, by (45) and (47), 



f a, = 2'" (m - ^)""' m ! P,,,, and 

 l(>„„^, = 2""+i(/« + i)<"'+i'm! A,«+i, 



and, because P^w ^"fl A-n+i ''^''*^ constant multiples of Q^.^ and ^2m+i) 



by (41), 



(50) TP,,„ = -m- P,,„ , TP,,„^, = - (m + ^) ■ P„„^^ , 



for all values of m. For the P's with the smaller suffixes we have, from 

 (48) and (7), 



(51) /^» = ^' ^i = -p^ = o, Ps = -yz, p. = -i + y,, 



I A = — 3 P3 + T^ P5 , A = — 2 + P4 — iV l>6 etc. ; 

 so that equations (50) give 

 (52) TPo = O-Po = 0, TP, = -iP, = 0, TP. = -Po_ = 0. 



Because Po„ is a linear function of the /j's with even suffixes up to 2 ?» 

 in which the coefficient of pgm is different from 0, and P^-i+i is a linear 

 function of the p's with odd suffixes up to 2 m + 1 in which the coefficient 

 of p2m+i is different from 0. p^m can be expressed as a linear function of 

 the -P's with even suffixes up to 2 m and pom+i as a linear function of the 

 P's with odd suihxes up to 2 m + 1. In fact, if we multiply Pj,,, or P,,,,^.! 



by ( — 1)'" ( ] and sum for all values of m from to k, we obtain 



PVc 



2^(/(,--i)<*'' 



k 



Si-'>"(»>-=S/-'> 



P2.+1 



2'>i(j + |)(m)^ 

 Pat+i 



^(-OC- 



2^+'(/l- + ^)'*+i'' 



