208 
DR. W. F. SHEPPARD ON 
q = l 
of the form 2 <p t (q) y q , where <f> t (g) is some polynomial in g of degree t. It follows 
9 = 0 
that any l.c. of <x 0 , o- ]5 < r 2 , ... cr t is also of this form. 
(v.) If we denote the f th moment of the y s by then Mf is of the form 
q = m 
2 <j> f (q ) y q . Hence cr t is a l.c. of M 0 , M x , M 2 , ... M t ; and M t is a l.c. of <r 0 , <r ]} cr 2 , ... <r t . 
9 = 0 
More generally, any l.c. of <r 0 , o- l5 a - 2 , ... «x t is a l.c. of Tf 0 , Hi), df 2 , ... df t ; and 
conversely. 
Reduction of Error (General). 
6. General Theorems .—-Let A, B, C, ... P, Q, R, ... be a set of quantities as in § 2, 
but all of the same kind. If 
w = aA + hB + cC +... (with or without terms in P, Q, R, ...), 
x = w+pP + qQ + rR+ ... , 
where a, b, c, ... are fixed and p, q, r, ... are arbitrary, and if we choose p, q, r, ... 
so as to make the m.s.e. of a? a minimum, the resulting value of x is called the 
improved value of w, using P, Q, R, ... as auxiliaries. The following are general 
theorems; some are quite elementary, but it is convenient to state them here. The 
specially important theorems are (III.) and (XIII.). The assumption mentioned 
under (VI.) should be noted. If strict proofs of (I.) and (II.) are required, the 
method should be that of Appendix I., § 2. 
(I.) The m.p.e. of A and any l.c. of A, B, C, ... is the same l.c. of the m.pp.e. of 
A and A, B, C, ... [i.e., m.p.e, of A and aA +bB + cC+... is a (A ; A)+b(A ; B) + 
c(A;C) +...]. 
(II.) The m.s.e. of any l.c. of A, B, C, ..., or the m.p.e. of any two such l.cc., is 
found by squaring the former or multiplying the latter and replacing squares and 
products by the corresponding m.ss.e. and m.pp.e. [i.e., m.s.e. of aA -\-bB + cC+... 
= a 2 (A ; A)+2ab(A ; B) + b 2 (R ; B)+2ac(A ; C)+2bc(B ; C)+c 2 (C; 0) + ..., and 
similarly for m.p.e. of a A + bB + cC+ ... and a' A + b'B + c'C+ ...]. 
(III.) If the improved value of A, using certain auxiliaries , is A + a, then the 
m.p.e. of A + a and each of the auxiliaries or a or any other l.c. of the auxiliaries is 
zero. [Let the auxiliaries be P, Q, R, ..., and let A+a = A+pP+qQ+rR+.... 
Then the m.s.e. of A + (p + 0) P + qQ + rR+ ... (= A+a + 6P) is (A+a;A+ a) + 
26 (A + a ; P)+0 2 (P ; P). In order that this may be a minimum for 0 = 0, 
(A+a ; P) must be zero. Similarly for (A + a ; Q), (A + a ; R), .... This proves the 
first part of the theorem ; the second then follows from (I.).] Hence 
(IV.) If the improved values of A and of B, using in each case the same set of 
auxiliaries, are A + a and B + /3, the m.pp.e. of A + a and, B + /3, of A + a and B, and 
of A and B + /3, are all equal; and 
