STORY. — A NEW GENERAL THEORY OE^ ERRORS. 177 



VI. Differential Equation of f(s). 



Formulae (22) and (24) give the variations of i/^ (i) and p^ due to the 

 addition of any infinitesimal source of error ; the mean s(|uare p^' of the 

 residual due to such a source is an essentially positive infinitesimal ; in 

 determining the mean powers of the residual for such a source by (13) 

 we may confine our attention to values of the residual that lie within 

 very narrow limits which, in accordance with the remark made in con- 

 nection with (15), include 0, so that we need consider only very small 

 values of ^; the terms of the integral (13) for greater values of k are 

 infinitesimals of a higher order than the corresponding terms of the 

 integral for smaller values of k, so that the integral ^j.' for any value of 

 k greater than 2 is an infinitesimal of a higher order than pj. AVe 

 specialize the infinitesimal source of error by assuming j)i.' — for 

 3 ^ k* "We have then, by (24), 



¥'« — ( •) ) P'"-'- ' P^' ^°^ '^ < '"' 

 from which, together with (6), follows 



p./ = 8pi = 2fj.- 8/z, 

 so that Sp„ = m (ni — 1) ■ p„._2M " V, 



while (22) gives 



We denote partial differentiation by the round 9, regarding i/^ as a 

 function of $, p^, Ps, Pij • • • and/ as a function of s, p^, pi, ps, . . . , so 

 that, by (7) and (27), 



9^_]_^ 92P_ I 9Y 



9$ ~ p.^ 9s 9e ~ ? 9s' ' 



The variations denoted by 8 were all taken for a defnite value of |; 

 therefore, by (7), (27), and the formulae just written, 



\p.J p. p, 



c'^ c-«' p. 



* Tills assumption does not in any way affect the generality of the results 

 obtained, because p^.' enters into \f> (^) only through the variations of the unac- 

 cented p's. 



VOL. XL. — 12 



