308 Mr. F. Y. Edgeworth on the Law of Error. 
So far we have taken for granted that the compound curve 
does tend to a limiting form, that the average curve is less 
dispersed than the original, on the strength of an unwritten 
common sense, or unsymbolic reasoning, such as Mr. Venn, with 
his usual clearness, has expressed in the chapter on the Method 
of Least Squares in his ' Logic of Chance.' But in Chance, as 
in other provinces of speculation which have been invaded by 
mathematics, common sense must yield to symbol. The fact 
1 c 
that in the case of one particular curve, — -. 2 - 2 , noticed by 
Poisson, the average is neither more nor less dispersed than, 
is the same as, the original, suggests that there may be forms 
on the other side, so to speak, of this particular form. And 
such, for instance, the form - (x positive both ways) 
is found to be; if the average of two curves be taken according 
to the principle of page 305 (modified for the case of odd 
powers of the variable). This property of divergence, as it 
may be termed, is even more clearly exhibited by the Laplace- 
Poisson analysis, if in our adumbration of that analysis 
(above, p. 307) we put t a positive fraction. h\ our second 
paper it will be shown that the advantage of taking an average 
depends upon the property that the integral \ u dx is for 
Jo 
every value of x greater in the case of the average than the 
original curve. This property is now seen not to be universal. 
With some surprise I find that divergence is not confined 
to curves all of whose mean powers are infinite. Take, for 
example, 
(l+.v)" 2 
The absolute term of the compound of two is, by our method, 
r i ^- i >% /-= 2 x (n - i)2 
J_„(l + *) 2 * 4 2n-l 4 
The absolute term of the average is double this; that is. less 
than the absolute term of the original. 
The preceding propositions have corollaries relating to the 
hyperphysical applications of the Calculus of Probabilities. 
(1) The conclusion that a considerable department of the 
Calculus of Probabilities, the calculation of probable errors, is 
less arithmetically precise than is commonly supposed, is, if 
correct, not unimportant; for the claims of the calculus to 
