254 DETERMINATION OF AX ORBIT FROM [BOOK II. 



determination of the unknown quantities will constitute a problem, indetermi 

 nate, determinate, or more than determinate, according as p&amp;lt;^v, [i =v, or 

 /j &amp;gt; v* We shall confine ourselves here to the last case, in which, evidently, an 

 exact representation of all the observations would only be possible when they 

 were all absolutely free from error. And since this cannot, in the nature of 

 things, happen, every system of values of the unknown quantities p, q, r, s, etc., 

 must be regarded as possible, which gives the values of the functions V M, 

 V - M , V&quot; M&quot;, etc., within the limits of the possible errors of observation ; 

 this, however, is not to be understood to imply that each one of these systems 

 would possess an equal degree of probability. 



Let us suppose, in the first place, the state of things in all the observations to 

 have been such, that there is no reason why we should suspect one to be less 

 exact than another, or that we are bound to regard errors of the same magnitude 

 as equally probable in all. Accordingly, the probability to be assigned to each 

 error A will be expressed by a function of A which we shall denote by (f A. Now 

 although we cannot precisely assign the form of this function, we can at least 

 affirm that its value should be a maximum for A = 0, equal, generally, for equal 

 opposite values of A, and should vanish, if, for A is taken the greatest error, or a 

 value greater than the greatest error: yd, therefore, would appropriately be re 

 ferred to the class of discontinuous functions, and if we undertake to substitute 

 any analytical function in the place of it for practical purposes, this must be of 

 such a form that it may converge to zero on both sides, asymptotically, as it were, 

 from A = 0, so that beyond this limit it can be regarded as actually vanishing. 

 Moreover, the probability that an error lies between the limits A and A -(- d A 

 differing from each other by the infinitely small difference d A, will be expressed 

 by (pJdJ; hence the probability generally, that the error lies between D and 



* If, in the third case, the functions V, V, V&quot; should be of such a nature that [i -j- 1 v of them, 

 or more, might be regarded as functions of the remainder, the problem would still be more than determi 

 nate with respect to these functions, but indeterminate with respect to the quantities p, q, r, s, etc. ; that 

 is to say, it would be impossible to determine the values of the latter, even if the values of the func 

 tions V, V, V&quot;, etc. should be given with absolute exactness : but we shall exclude this case from our 

 discussion. 



