184 
ME. M. W. CEOFTON ON THE PEOOE OF 
diminutive Error may be rejected in comparison with its mean square*. We infer, 
therefore, 
If y=f{x) represent any Error of indefinite amplitude , and if a new Error , y—$(x), 
of indefinitely small importance as compared with it, be superposed, the resulting compound 
Error will be represented by the equation 
y =/0 ) - “ e/W + ^ 
( 9 ) 
where a, X are infinitesimal constants, viz. the mean value of the new Error and the 
mean value of its square f, the number of observations being supposed the same for the 
Error (9) as for the former, y=f(x). 
If we now conceive y=<p(x) in the above to be one of a large number of independent 
infinitesimal Errors, and y =f(%) to be the compound finite Error resulting from the 
combination of all the others, we infer from (9) that each elementary Error yz=<p(x) 
affects the law of the combined Errors in a manner which only involves (a) the mean 
value of the elementary Error, and (7.) its mean square. But if this be so, we may, for 
our present purpose, substitute for y=q>(x) any other Error whatever which has the 
same mean value and mean square (provided of course its mean cube &c. may be neg- 
lected in comparison with its mean square). We may therefore for our purpose replace 
y=<p(x) b y+ 
fl (if— q) “ 
y= vmr=W) e (1 °) 
which fulfils these conditions. 
Likewise, if there be another elementary Error whose mean value is (3 and mean square 
f L , we may replace it by 
11 VZTlfi-fi) 
e ^- 00 , 
and so on, for all the elementary Errors. 
of any number of them will be 
N 
^ 2ir{\ + ju. + v + . . . . — a 2 
Hence (see equation 8) the Error compounded 
(x—a—P— y—. . . .)f 
— ■ ■ ■ - - p 2 (A + /x + . . . . — a 2 — 8' 2 — . . .) 
-/3 2 -y 2 -...) 
* We cannot neglect the mean square as compared with the mean 1st power, as the latter is the algebraical 
sum of a number of positive and negative elements, which sum may he of any amount, however small, and may 
sometimes vanish altogether ; whereas the former is the sum of a number of positive elements, and therefore 
cannot vanish. 
f If the new Error he such as to give any discontinuous distribution of points (see art. 4), it is easy to satisfy 
ourselves, by the method of art. 6, that the above proposition still holds good. In fact, if the n values of the 
new Error be x 2 , x 3 , . . . ., we shall have, instead of the formula (7), 
n^xM-xd+fit- O +M ~ »,) + &c. }, 
which is true in all cases, whether the distribution be continuous or discontinuous, or a mixture of both ; and 
hence the formula (9) will follow. 
1 This suggestion is due to Professor J. C. Adams, one of the Eeferees charged by the Eoyal Society with the 
duty of reporting upon the present Paper. The remainder of the proof, which was of a different nature in the 
Paper as originally presented, is much simplified thereby. 
