STORY. — A NEW GENERAL THEORY OF ERRORS. 181 



any odd suflix into a linear function of tlie ps with odd sufRxes. There- 

 fore, if H is a linear function of f)2,„ and the p's with even suffixes less 



1 than 2 m, wliere 2 ^ m, ( T + m) u does not involve pom' {T -{• my^hi 

 involves neither po,,. "or po,„_., (7'+ m)'"' ?< involves neither p^,,,, po„,_2, 

 nor p2m-4, a"'^ •'*o on, and ( T + w)""""?* is a constant (or linear function 

 of po and po) ; therefore, by (3G) 



(39) ] V / V y 



I =(T+ 1) T- (T+ mY"'-^^ u = 0; 



that is, (7^-f J^i)'"'"*"^' annihilates every linear function of the p's with 

 even suffixes not greater than 2 m, where m is any positive integer or 

 (this is true for m = or 1 because the operator contains T'as a factor). 

 But (T -{■ w)'"'+^* annihilates no linear function that involves a p with 

 an even sutfix greater than 2 m or an odd suffix as great as 3 ; namely 

 if Pi- is the p with greatest even suffix or greatest odd suffix (other 

 than 1) that is involved in the linear function u (with a coefficient dif- 

 ferent from 0), the coefficient of p^. in (T + my'"'^^^ u is the coefficient 



of Pi- in u multiplied by { "^ — 7^] ' whose factors are all negative if 



k is an even number greater than 2 m and all odd multiples of ^ if k 

 is odd. 



Similarly, if n is a linear function of p^m+i and the p's with odd suffixes 

 less than 2 m + 1, where ^ m, 



. (40) (T+ m + -i)""+^'« = 0, 



so that (T + m + -j)'"'"*"^* annihilates any linear function of the p's with 

 odd suffixes nut greater than 2 ?n + 1 ; but (T -\- m + •j)'"'"'"" annihilates 

 no linear function that involves a p with an odd suffix greater than 



2 m + 1 or an even suffix. 



Applying (39) and (40) to p2,„ and p2,„+i, respectively, we have 



(T+ /;0""+" P2. = 0, {T+ m + i)""^"p2,„+i = 0, 

 so that, by (34), 



r(r+ m - l)""'p2,,. = - m • (r + m - l)""'p2»., 



T{T+ m - i)"'"p2„.+i = - ('« + h-{T-\- m- i)<""p2.+i, 

 or 



(41) T Q,„ = -m- Q,„„ T Q,„^, = - (m + i) ■ Q,„,^^, 



