216 
DR. W. F. SHEPPARD ON 
the values of the n’s when we fit by least squares are given (“ Fitting,” §§ 1 , 2 ) by the 
equations (f = 0 , 1 , 2 , ... j) 
Xgfi a 0 + Zg /+1 . a^ + ... + 'Eq f+j . cq = 'Zq f u q = My.(55) 
7 7 9.7 
These are the same equations that are given by the method of moments. 
(ii.) The above equation (55) is a statement that the f th moment of the vs is equal 
to that of the us. In order to prove that the process of reduction of error, using 
differences of order exceeding j as auxiliaries, gives the same result, it is sufficient to 
show (a) that the improved value of u q as given by this process is of the form of v q 
in (54), and (0) that the f th moment of the improved values of the us is equal to 
that of the original values for f = 0 , 1 , 2 , ...j. 
(iii.) We have shown, in §11 (ii.), that the improved value of u q is a polynomial of 
degree^’ in q. This establishes (a). 
(iv.) By (XIII.), the ( /’ th moment of the improved values of the us is equal to the 
improved value of their f th moment. In order to show that this is equal to the 
original value of the f th moment, it is sufficient, by (IX.), to show that the m.p.e. of 
the original f th moment and every difference of order exceeding j is zero. 
(v.) Let the k th difference of u,._ k be 
4 = K U r -k 1 u r _ 1 + ... + {-)%u r _ k . 
Then the f th moment is 
... +r- f u r +(r—iyu r _ 1 + (r—2yu r _2 + 
and the m.p.e. of this and 4 is 
k 0 r / —k l (r—l) f +k 2 (r—2) f — .... 
But this is the k th difference of (r—k) f , and is = 0 if & > f 
This proves the proposition. 
13. Fitting by Least Squares .— Next suppose that the set is not self-conjugate. 
If the Si’s were the differences of a set of us, we should fit a polynomial of degree 
(say) j to the u s. This suggests that, in the more general case, the u’s being 
connected with the <fs, as in § 10 , by the relation ( r — 0 , 1 , 2, ... 1) 
u r = ( r o) 4 + ( r i) 4 + ( r 2) 4+... +(r z ) 4>. (56) 
we should try to fit an expression of the form 
v r = (r 0 )e 0 + (r 1 )e 1 + (r 2 )e 2 +...+(r j )e j .( 57 ) 
to the us by an appropriate method of least squares ; the (r)’s being the same as in 
(56), and the e’s being the quantities to be determined. 
