PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 141 



of any combination whatever of independent events, and consider the deviation of 

 this combination from the most probable combination as the Error, and the ratio of 

 its probability to that of the most probable combination, as the function which 

 expresses the Law of Error. If we find, as we proceed, that the law thus arrived 

 at, is (in form at least) totally independent of the number, variety, &c, of the 

 several simultaneously acting causes, we shall thus have a very strong argument 

 in favour of the correctness of the process ; whose real difficulty, be it remembered, 

 is logical and not mathematical. The mathematical processes to be employed 

 below are, of course, known, and will be found in most treatises on Algebra ; 

 but, for the present application, it will be convenient to put them in a form 

 slightly different from the usual one. 



8. Taking the simplest case, let us suppose a bag to contain white and black 

 balls, whose numbers are as p : q, where p + q = l. The chance of drawing a 

 white, and (3 black, balls in n ( = a + /3) drawings, replacing before each drawing, 

 and disregarding the order in which they appear, is, — 



v*<f (i) 



13 



This is a maximum, when a : j3 : : p : q; which, when n is indefinitely great, can 

 always be exactly attained. This maximum value is, — 



\pn \qn 



The ratio of these two numbers is, — 



p l " l q^ (2) 



&M P -p*<r*> ( 3) 



Now, according to the principle above assumed, we must treat a —pn, the devia- 

 tion from the most probable result, as measuring the error in some observation, 

 while the expression (3) measures the probability of it, as compared with that of 

 the most probable result. To introduce the ordinary notation, let x be the error, 

 and y the (indefinitely small) probability of that error ; then, A and m being 

 constants, — 



a.—pn = mw, ...... (4) 



while y may be expressed as the product of (3) into A, that is, by (4), 



y = A , I— |— p mx q- mx ... (5) 



\pn-\-mx \qn — m.r 



When n is a large number, the value of this is easily found from Stirling's 

 Theorem, viz.— 



1.2.3 . . . . n = |n 



= ^27r^ + V"(l + T ^+&c.) 



