5^ Prof. Donkiii on certain Questions rclutiny to 



If we arc asked what is the method of obtaining; the most pro- 

 bable result from a system of observations not numerous enough 

 to justify, as an ai)})rox.huation, the supposition that they are 

 injinite in number, it is plain that no answer can be given till 

 we are told whether it is to be assumed that the law or laws of 

 facility of errors in the individual observations are known, or 

 unknown ; or, to speak more accurately, until we are told what 

 is to be assumed as the state of information of the observer con- 

 cerning the laws in question. For the probability of every 

 hypothesis depends upon, the state of information presupposed 

 concerning it. 



If the law of facility of eiTors (which we will suppose, for sim- 

 plicity, the same in all the observations) be assumed as known, 

 the problem involves no difficulty of principle, though for most 

 laws the required integrations would be impracticable. 



But if the law be wholly or partially unknown, though it is 

 still easy to indicate the way in which the problem ought, theo- 

 retically, to be treated, the processes required are, in all actual 

 cases, entirely beyond the present powers of analysis. 



To illustrate this, consider the case in which all the observa- 

 tions refer directly to a single unknown quantity x. If a, a', 

 fl", ... be the obseiTcd values, and </> were known to be the func- 

 tion expressing the law of facility of errors, then the probability 

 that the true value lies between x and x + dx would be 



Q.(f>{x-a)(f>{x'-(^)4>[x-'a^^) ,,.dx, . . . (1.) 



where C is determined by the condition that the integral of this 

 expression, extended to all admissible values of x, shall be 

 equal to 1. 



Now suppose that the function <f> is not known, but may be 

 of any of the forms <j)^, (fy^, <l>3, - - - and let pi be the probability 

 that it is <^^.. Then instead of the expression (1.) we should have 



X{C.p.(t).{x-a)(P,{x-a') .... dx}, 



the summation extending to all the actual values of i . 



In the ordinary cases occurring in practice, nothing is known 

 of the form of (f), except that it must satisfy some veiy general 

 conditions, such as that smaller errors are more probable than 

 larger, &e. ; the number of supposable forms is therefore infinite, 

 and the summation indicated in the preceding expression would 

 depend upon a calculus bearing the same relation to continuous 

 variation of /or?/?, that the integral calculus does to continuous 

 variation of value. Such a method, it is needless to say, does 

 not at present exist ; the calculus of Variations being, with refer- 

 ence to functional fo?'m, the imperfect analogue, not of the inte- 

 gral, but of the differential calculus. 



