THE LAW OF ERRORS OF OBSERVATIONS. 
183 
where 
T* a / ^ a f* a 
! x<p(x)dx i x~<p(x)dx \ x 3 <p(x)dx 
—-La , x—'Li , <t = La &• 
I <p(x)dx \ <p(x)dx \ <p(x)dx 
J-b J-6 J-i 
, &c., 
a being the mean value of the Error y=p(x), X its mean square, g its mean cube, &c. 
7. In the problem of finding the law of error resulting from the superposition of a 
great number of Errors, each of very small importance by itself, we will consider each 
component Error as the diminutive of some Error of finite importance* (see art. 5). 
Thus if y — E(x) be some possible finite Error, and we reduce its dimensions in the 
U / tV\ 
ratio i, where i is infinitesimal, the diminished Error will be |=FL ) ; and if the 
mean value, mean square, mean cube, &c. of the former be called 
E 1? E 2 , E 3 
? 
it is easy to see that the same means, for the reduced Error, will be 
«Ej, i 2 E 2 , ^ 3 E 3 , 
Now adopting the usual axiom that no function can represent a finite Error unless 
E 1? E 2 , E 3 , . . . . are finite , it follows that the mean cube, mean 4th power, &c. of the 
* Thus all conceivable cases of Errors whose extreme limits, or amplitude, are very small, are contained in 
the above method of proof ; also those small Errors which, though their extreme amplitude he not very small, are 
merely possible finite Errors (of great or infinite amplitude) on a reduced scale. It is necessary, however, to 
observe, in examining the nature of all the minute simple Errors which our hypothesis in its generality com- 
prises, that there are cases quite conceivable, and involving no absurdity, of simple Errors of trivial or infini- 
tesimal importance which come under neither of these categories, and to which the method in the text will 
not apply. To give a simple instance, imagine an occasional source of Error, which rarely operates, but which, 
when it does, gives a fixed finite error lc (thus we may conceive an observer to mistake, once in a thousand 
times, the succeeding division of his instrument for the true one). Let this happen on an average once for n 
times that the cause is not in operation ( n being supposed very great) ; then the mean value of the Error is 
lc . lc 2 
, its mean square is , &c. It is therefore of infinitesimal importance whether, with Laplace, we 
n + 1 n + 1 
estimate the importance of an Error by its mean value (irrespective of sign), or, with Gauss, by its mean square ; 
but as its mean cube &c. cannot be rejected in comparison with the mean square, the above analysis cannot be 
applied to it. Minute simple Errors of such a description must then be excepted from those which are supposed 
to enter into the composition of the actual errors of observations. If an appreciable number of them did enter, 
the received exponential law could not hold for the compound Error. Thus were we to combine a large number 
of small Errors of the nature of the simple instance just cited, the resultant Error would be of a discontinuous 
nature, represented by groups of coincident points, with finite intervals between them. 
Though it is necessary clearly to understand that the full generality of the hypothesis is restricted by the 
exceptions explained in this note, yet there seems every reason to suppose that such cases are too rare in practice 
to cause any sensible deviation from the exponential law of error, the great majority of the minute component 
Errors which jointly affect any observation in rerum naturd having each, it is natural to suppose, a very minute 
range or amplitude. 
2 b 2 
