STORY. — A NEW GENERAL THEORY OP ERRORS. 191 



P^ =: p., = 0], — iu fact, they have been assumed to be entirely arbi- 

 trary ; therefore, the first of equations (79) for < m < a and the 

 second of these eijuations for < m ^ ^ must be satisfied independently 

 of the values of the P's involved ; that is, for any one of the values of m 

 in question, the coefficient of any P in the right member of the corre- 

 sponding ecpiation must be excepting the coefficient of the P that 

 occurs in the left member, which must be 1. Therefore, 



S-2m ^2m ds = 1 for < m < a, excepting tn = 1 ; 



(SO) 



(81) 



0.2,„+i j S.2„,+i i?.2,„+i ■ ds=l for 1 < w ^ ^ ; 



s 

 s 



I -S-zm ^ik ' ds = for ^ m "^ a, ^ k "^ a, k ^ m ; 



I Sn^Rok+i ' ds = for < m ^ a, ^ ^ ^ ^ ; 



7 

 fs.2,„+i R.,-ds = for ^ m ^ /3, ^ ^ ^ a ; 



/ 'S'Zm+i R'ik+i ' ds — for ^ m ^ f3, <:^ k ^ (3, k ^ m. 



s 



From the first, second, third, and fourth sets of equations (81) we ought 

 to exclude cases for which ^- = 1, 0, 1, and 0, respectively, because 

 Pi = P2 = and the terms corresponding to these cases do not occur in 

 (79) ; but, by (68), 



(82) Cs„, B,ds = fs,, S - e~'^ ■ ds = Cs, R„ • ds, 



so that tlie cases that are to be excluded from any one set of equations 

 (81) will occur (with an interchange of »i and k) in other sets, excepting 

 only the case of the integral 



