THE LAW OE EKROES OF OBSERVATIONS. 
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this absolutely ; thus it is not enough for our purpose to show, could we do so conclusively, 
that each error in practice is compounded of a large number of smaller errors ; we must 
also show that they are independent , at least for the most part. Thus we may conceive 
one of the minute errors affecting an astronomical magnitude to be an error in the 
refraction proceeding from a rise in the general temperature, and another affecting the 
same observation to be an error of time arising from the expansion of the pendulum 
through the same cause ; now these two minute errors are not independent, and would 
have to be mathematically combined in quite a different way from two that were inde- 
pendent ; and, indeed, such a change of temperature would influence the actual error of 
the observation in other ways also. However, we may at least safely conclude that the 
hypothesis in question is not a mere arbitrary assumption, but a reasonable and probable 
account of what does in fact take place in the case of careful and refined observations. 
3. In proceeding to submit this hypothesis to mathematical analysis, the minute simple 
errors which go to form the observed compound error will be assumed to follow each 
its own unknown law, expressed by different unknown functions of the utmost generality*: 
positive and negative values of each error will not be assumed equally possible ; on the 
contrary, the cases will be included, as obviously ought to be done, of minute disturbing 
influences which always cause the observed magnitude to err in excess, and of others 
which cause it to err only in defect. I will exclude all mention of the term probability, 
and will consider solely the frequency or density of the error, viewed as a function of its 
magnitude. 
Let any magnitude which has to be determined by observations affected with some 
one cause of error (simple or compound) be represented by the line BA ; 
let a large number of such observations be made, and let the observed values be repre- 
sented by a number of lengths BA', measured from B : it will be found in general that 
in the neighbourhood of A the line will be dotted over with a multitude of points A', 
the distance AA' being the error in each case. These dots will begin at some point C, 
and end at some point D, which generally are on opposite sides of A, but may both be 
at the same side. Between C and D the dots will be distributed over CD with a varia- 
ble density : this density, at any point A', will represent the frequency or density of errors 
of magnitude AA'. 
If at every point A' we erect an ordinate A'P representing the density at A', we shall 
thus trace out a locus or curve C'D', whose equation we may call, taking A as origin, 
* With regard to the limits or amplitudes of the errors, see note on art. 7. 
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