AND THEORY OF ERRORS. 25 



(67) 



R = C Z7i ]n - C U^ \n e~ k fl + k + ~ + . . . + r^jY 



We may determine the value of the constant U^ by the con- 

 dition that R = 1 for k = oo. This gives (7 7^ jw == 1, and 



K = l _ e - k (l + A; + ^ . . . + [^ZTfV W 



^ 



(69) 



It is worthy of notice that the form of these equations de- 

 pends only on the number of degrees of freedom of the system, 

 being in other respects independent of its dynamical nature, 

 except that the forces must be functions of the coordinates 

 either alone or with the time. 



If we write 



** 



for the value of k which substituted in equation (68) will give 

 R = 1, the phases determined by the equation 



F--=k B= i (70) 



will have the following properties. 



The probability that the phase falls within the limits formed 

 by these phases is greater than the probability that it falls 

 within any other limits enclosing an equal extension-in-phase. 

 It is equal to the probability that the phase falls without the 

 same limits. 



These properties are analogous to those which in the theory 

 of errors in the determination of a single quantity belong to 

 values expressed by A a, when A is the most probable 

 value, and a the 'probable error.' 



