STORY. — A NEW GENERAL THEORY OF ERRORS, 185 



and, evideutly 



(58) G{T)- P.,"' P,"* P/-' . . . = G^ ("- ^y P3"' P,"* A"' . . . 



where G (T) is any polynomial in T. jMore generally, if 11 is any 

 isobaric polynomial of weight w in the P's, that is, any polynomial whose 

 terms are all of one weight w, 



(59) G{T)-ll^ (?(^_«'y ri. 



Therefore, if the coefficients of 11 are not all 0, G (T) • 11 is or is not 



according as G (T) does or does not contain T -^ — as a factor. In 



particular, (T + m)'"'+^' annihilates every polynomial in the P's whose 

 terms are all of even weights not greater than 2 m and annihilates no 

 polynomial of other description, while {T+ m + |^)''"+i* annihilates every 

 polynomial whose terms are all of odd weights not greater than 2m + 1 

 and no others. By means of formula (59) any polynomial in T can be 

 applied to any polynomial in the P's ; namely, any polynomial in the 

 P's can be expressed as the sum of isobaric polynomials of different 

 weights and the polynomial in T can be applied to each isobaric part 

 by (59). 



The P's can be determined from the function y(s) ; namely, from (7), 

 (13), and (27) we have 



f . 1 



(60) p, = ^l ^ /^.lA(^) • d$ = fs' -/(s) ' ds for ^ k, 



i ^ 



and obtain, by the substitution of these values of the p's in (48), 



s .s 



(61) P,,„ = Js„„ •/(«) • <h, P,„,^, = Js,„,^, -/(s) . ds, 

 where 



(62) 



--!,(-)■ COW 



m 



i)"' 



(i + i)"+l' 



for < m. For the S's with the smaller suffixes we have 



