Mr. F. Y. Edgeworth on the Law of Error. 307 
(analogous to that of infinite mean powers) which, though 
postulated by theory, are absent in fact. For instance, the 
mean exponentials of the form 1 e +xA x </> dx, where <£ is the 
%) — oo 
law of error, ought to be infinite for the probability-curve. 
But in fact a finite cluster of points cannot present this infinity. 
(3) The objection maybe transferred from experience about 
(presumably) compound errors to assumption about elemental 
errors. It seems sufficient to reply that, as the probable error 
of a non-exponential facility-function may be made infini- 
tesimal, so an element of this type may become identical with 
the received type to all intents and purposes, except indeed 
that of generation. The body of the two forms may be sen- 
sibly identical ; what shape is to be (approximately) assigned 
to the tail is a nice question of relative infinities, concerning 
which, I suspect, not much is known. 
(4) In fine it may be objected that the non-exponential 
curves are not reproductive in the same sense as the proba- 
bility-curve : that the superposition of two curves of the 
former kind does not result in a curve of the same type. 
"Whatever force there is in this objection is counteracted by 
the remark that the property in question is shared by multi- 
tudes of forms other than the simple exponential — namely, all 
of the type 
f(x) = — I e _a cos a.x . da (a not changing sign), 
where t is any positive quantity, integer or fractional. For 
1 f x 1 t 
— I f(x) cos ax dx = — e _a . 
Whence it appears, by putting « = 0, that f(x) fulfils the con- 
dition that its integral between the extreme limits should be 
unitv. And it also fulfils the condition that -J- should be 
dx 
always negative. Xow, by the Laplace-Poisson analysis, the 
compound curve is found to be 
* J -co 
COS ax da, 
which, by putting a =s t ft, we may reduce to the parent 
type. The probable error of the average is that of the original 
curve divided by s 1-- ^; not in general s*. Which was 
especially to be proved. 
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