(31) 



STORY. — A NEW GENERAL THEORY OP ERRORS. 179 



{T{u + v) = Tu+ Tv, 

 T- Cu= C- Tu, 

 T- uv = u- Tv + v Tu. 



If n is a function of the p's with sulFixes not greater than m, Tu is also 

 a function of the p's with suffixes not greater than m; if 2< is a poly- 

 nomial in the p's, Tu \^ sl polynomial in the p's of the same degree as u ; 

 in particular, if u is a linear function of the ps, with suffixes not greater 

 than m, J'm is a linear function of the p's with suffixes not greater than 

 m. If 3 ^ m, 



(32) 7V,„ - \ m (m - 1) • p,„_2 — }. m ■ p„.. 



Therefore, if u is a linear function of p^ and the p's with suffixes less 

 than m, where 3 ^ m, T" u is a linear function of p„, and the p's with 

 suffixes less than m in which the coefficient of p„ is the coefficient of p,,^ 



in u multiplied by I — — ) , where n is any positive integer or 0; more 



generally, if G(T) denotes any polynomial in T (linear function of 

 powers of T with constant coefficients), G (T) • u is a linear function 

 of the p's with suffixes not greater than m in which the coefficient of 



p„ is the coefficient of p„, in u multiplied by G i ] ; in particular, 



G{T^ • u will not involve p„, if G (T) contains 7^+ — as a factor, and 

 only then. Formula (32) holds also for m = 0, 1, and 2, giving 



Tpo = • po = 0, Tpi = — 1 pi = 0, Tp2 = po — p2 = 0, 

 because po = pa = 1 and pi = 0, by (7) ; therefore, 



(T+ §) Po = 0, (T+ 1) p, = 0, (r+ f) p, = p, = l', 



so that G(T)' p^ does not involve p,„ if G (T) contains the factor 



tn 

 T + —, where tn is or any positive integer excepting 2, and G(T) • p„ 



does not involve p2 if G (T^ contains J" as a factor. Of course, poly- 

 nomials in T are commutative with each other, so that it is of no conse- 

 quence what position in a product G (T) the annihilating factor occupies. 



It is convenient to employ the usual symbol y"' to denote the product 

 of ?i factors diminishing by successive units from y, whatever meaning y 

 may have (it may be a symbol of quantity or of operation), so that 



(33) y" = y (y - 1) (y _ 2) (^ - 3) . . . (y - n + 1) 



