180 
ME. M. W. CEOETON ON THE PEOOE OF 
This we may call the curve ox f unction of Error *. It is of course generally discontinuous, 
as it is only to include values of x between the points C, D. The function <p(x) strictly 
speaking should vanish for all values of x beyond C and D ; however, we shall not require 
any consideration of the analytical methods of expressing such functions. If N be the 
number of observations taken, and if we put AD=a, AC=6, then as ydx denotes the 
number of errors lying between x and x-\-dx, 
N= f <p(x)dx (2) 
J ~ b 
It is well to notice that, if C be any constant, the equation 
y=C<p(x) 
really is the same function of Error as (1), the number of observations only being altered. 
4. In order not unduly to limit the generality of the investigation, it is necessary 
further to study the nature of the possible ways in which the dots we have spoken of as 
representing the observations may be scattered along the line CD, in the case of various 
unknown simple causes of error ; noting also what becomes of the function <p(x) and the 
curve C'D' in each case. And first, in many cases the dots will be distributed conti- 
nuously along CD, thus giving a curve without gaps or intervals. It is by no means 
necessary that this curve should descend towards CD at its two extremities more than in 
the middle ; in other words, the extreme values of a simple Error are not always less 
probable than the intermediate ones. There may be cases where the extreme values are 
the most probable ; for instance, the Error occasioned by supposing a point fixed, which 
is in reality performing extremely minute and slow oscillations about its mean position. 
But besides the cases of continuous distribution, there are others, not only conceivable, 
but which we may be sure do actually occur, in which a function or curve does not 
assist our conceptions, and we shall do better merely to consider the points or dots 
themselves. There may be what is called a constant Error ; that is, some cause which 
gives the observation always too great (or too small) by the same fixed minute amount : 
the distribution here is simply a group of N coincident points somewhere on CD. Or a 
certain cause may only admit of two or more definite values for the error ; the distribu- 
tion will be two or more groups of coincident points, the numbers in each group 
being equal or unequal. Again, an important class of Errors are those which may be 
called occasional Errors , that is, produced by intermittent causes not always in operation. 
In such a case, if N observations be made, a certain number of them (say n) are unaffected 
by the Error ; the remaining N — n, made when the cause is in operation, we may suppose 
represented by dots continuously or discontinuously distributed ; we have then a group 
of n coincident points at A, besides a number N— n distributed in some way over CD. 
Errors of mistake or forgetfulness, and many others also, are of this description. 
* The word “ error ” is sometimes used for shortness to express a source of error. To avoid confusion we 
may write it with a capital E, when used in this sense. Thus “an Error” will mean a source of error, or the 
assemblage of actual errors (or the curve or function symbolizing them) which that source produces in a large 
number of trials, and which form a visible manifestation or representation of it : “ an error ” will mean a par- 
ticular magnitude. 
