STORY. — A NEW GENERAL THEORY OF ERRORS. 187 



the true form of $ (s), even if its complete expression is an infinite series, 

 tlie more terms of (GG) we take into account. Taking any finite num- 

 ber of terms, say for values of i up to /c, substituting in (64) and the 

 resulting expression fory(s) in (Gl), we obtain 







where 



s' Ji„, ' ds, 



s- 



(QS) J?„. = ^„ • e" 2 for ^ m. 



We see from (67) that, to the degree of approximation that corresponds 

 to K + 1 terms of (66), the P's are linear functions of the k + 1 ^'s, 

 whose coeflicients are definite functions of the limits s and s alone, because 

 the *S"s and, therefore, the ^'s are independent of the P's. Therefore, 

 not more than /< + 1 of the P's (including Pq, Pi, and P..) are linearly 

 independent ; the solution of such equations (67) as correspond to linearly 

 independent P's gives the k + 1 ^'s as linear functions of these P's, 

 whose coefficients are definite functions of s and s derived from the inte- 

 grals involved in (67) ; these expressions of the ^'s will involve a certain 

 number of arbitrary constants if the number of linearly independent P's 

 is less than k -\- 1, but that is of no consequence. If the P's are sub- 

 jected to infinitesimal variations, — say 8P„ is the variation of P„,, — 

 the ^'s, ^(s), and 6 (s) will undergo variations, which may be repre- 

 sented by 8A, (for A,), 8/(s), and S6(s), respectively. Then, by (66) 

 and (64), 



^0 (s) =S ^ s\ 8/(5) = 80 (s) - e-¥' 



and, by (Gl) and (G8), becausey(5) = for each of the limits s and s, 



F J J 



8P,„ = Is,, • S/(s) • ds = Cr^ ■ se (s) ■ ds =^ ~ Cs^ R,„ ■ ds ; 



that is, the relations between the 8P's and the 8^'s are the same as if 

 the coefficients of the ^'s in (67) were constants (whether s and s are 

 functions of the P's or not) ; therefore, in determining the ^'s as func- 

 tions of the P's that satisfy the differential equation (65) we may regard 

 them as linear functions of the P's with constant coefficients. The 

 necessary and sufficient conditions for these linear functions are those 

 implied in (Go) together with 



