192 REPORT— 1888. 



several years' experience, and roughly applicable to the unexamined data of any 

 year. 



(a) In the former case there is nothing more to be said, except that it will be 

 legitimate in the evaluation of the modulus to put for w^, w.,, &c., their apparent 

 values ; which may be written w, + A2v^, w,, + Aw.,, &c. For the error thus 

 introduced into the modulus is of a neglectible order. 



((3) The general expression in terms of the E-fluctuation is found by consider- 

 ing that the most probable value of the quantity under the radical sign in the last 



written expression is a/Cw/ + ^v■i^ + &c.) ^ » ^^^ere — is the mean square of error 

 measured, not, as before, from the simple (arithmetical) mean (of many batches of 

 ^'s), but from the weighted mean ^^ ; a difference which may easily be shown to- 

 be for our purpose of an order which may be neglected. 



This may be proved thus : 



The deviation of any p from the Weighted Mean — the relative or proportionate- 

 deviation — E' 



Sp2U 

 sin ^ '* 



Spw 



SIV 



This ratio may be thus expressed in terms of E,, the deviation of pr from the 

 Simple Arithmetic Mean. Put v for the difference between the weighted and 

 simple means. Then we have 



S;, 



1 " 

 — u i — - 



n p 



if we put p for the Arithmetic Mean of the ^'s. 



Sp Sjojo py +p.^ + &c. p^tVy +Pi^i + &t:. 

 JNow v= ,j — gj„ - ,j - Wi + t<;2 + &c. 



Substitute for^, its value j:> (1 + Ej) (where p is the Arithmetic Mean of the- 

 p's) ; and we have 



[E, + E2 + &c. _ WxE, + w „E„ + &c. ~j 

 n v>^ + W2 + &c. J 



Sw Siy 



1 ' w, 2 — — w., 



= - »Ei 2L. — + - pE„ — — : + &c. 



n n 



Put for the relative deviation of any w from the Arithmetic Mean of all the ?f;'s 

 (the coefficient of - p^r in the last written expression) r)^. Then we have 



v = ~ p [Ei»;, + E.,i?., + &c.]. 

 n 



The expression in brackets hovers about the value zero according to a law of error 



whose modulus is — ^-^r-; where C, as before, is the modulus of the E's, and 5" is 

 V 2 



the mean square of the »>'s. Hence - is of an order a/^ times smaller than Cx- 



But from the equation connecting E' and E it appears that the sum of squares 

 E'l^ + E'2* + &c. which occurs in the complete expression for the modulus of 



may be written 



