220 
DR. W. F. SHEPPARD ON 
same result, they are essentially different. This is explained in § 22 of “ Reduction.” 
The main difference may be expressed as follows :— 
(i.) In “fitting” we deal directly with the particular case. We assume that the 
true values follow a specified law, involving unknown constants, and we deduce 
values for these constants from the data by the principle of inverse probability. 
(ii.) In “reduction of error” we do not use inverse probability, and we only deal 
incidentally with the particular case. We regard the aggregate of the data as one of 
an indefinitely great number of possible aggregates from the same true values, and 
we use a method which will reduce as much as possible the m.s.e. of these possible 
aggregates. 
Some Steps in the Genekal Solution. 
17. Preliminary. — (i.) Our object is to find the improved value of any l.c. of the 
<fs or the u s, and the m.s.e. of this improved value or the m.p.e. of two improved 
values. Ordinarily, as already stated in § 8 (iv.), the quantity to be improved would 
be expressible in terms of the d’s, so that we need consider only the improved values 
of the S’s, i.e., the e’s. There are then four problems before us, viz. : (l) expression 
of the e’s in terms of the <fs; (2) expression in terms of the o-’s ; (3) expression in 
terms of the y s; (4) determination of the X’s. For practical purposes (2) is more 
important than (l) or (3), since there will be fewer coefficients involved. 
(ii.) Although it does not seem possible to obtain a general solution, otherwise 
than by determinants, there are some general propositions that indicate stages in the 
solution. If, without necessarily finding the complete expressions of the e’s in terms 
of the cfs, we can find for each e the coefficient of the first of the auxiliaries, then it 
will be seen from § 18 that we can find all the e’s if we know the E’s, and from 
§ 19 (i.) that we can find all the X’s if we know the A’s. It follows from (40) that in 
this latter case we can at once obtain the e’s in terms of the As. 
(iii.) We use the notation of § 7, and we also write 
— Ofj EE coefficient of <5) (as auxiliary) in (ey) ; -_ 1} 
so that 
Sf.j = o if/>i.(74) 
6 j,i = 1.(75) 
It should be observed that 64 is not equal to 64 The 0 ’s may be known directly, 
or, as is shown in (83), we may be able to obtain them from certain of the X’s. 
18. Formula of Progression. —The quantities which we want to find are the 
improved values 
of 4 using 4 4 S 3 , , 
of S } and 4 using 4, 4 S 4 , ... , 
of 4 4 and 4 using 4 4 4 ■ • • «. 
