AND THEORY OF ERRORS. 23 



...+i^ (?."-<?."), 



(55) 



and as in IF" terms of higher degree than the second are to be 

 neglected, these equations may be considered accurate for the 

 purpose of the transformation required. Since by equation 

 (33) the eliminant of these equations has the value unity, 

 the discriminant of F" will be equal to that of F 1 , as has 

 already appeared from the consideration of the principle of 

 conservation of probability of phase, which is, in fact, essen- 

 tially the same as that expressed by equation (33). 

 At the time t\ the phases satisfying the equation 



F' = k, (56) 



where 7c is any positive constant, have the probability-coeffi- 

 cient C e~ k . At the time tf", the corresponding phases satisfy 

 the equation 



F" = k 9 (57) 



and have the same probability-coefficient. So also the phases 

 within the limits given by one or the other of these equations 

 are corresponding phases, and have probability-coefficients 

 greater than C ' e~ k , while phases without these limits have less 

 probability-coefficients. The probability that the phase at 

 the time t f falls within the limits F' Jc is the same as the 

 probability that it falls within the limits F" = k at the time t", 

 since either event necessitates the other. This probability 

 may be evaluated as follows. We may omit the accents, as 

 we need only consider a single time. Let us denote the ex- 

 tension-in-phase within the limits F = k by Z7, and the prob- 

 ability that the phase falls within these limits by R, also the 

 extension-in-phase within the limits F = 1 by U r We have 

 then by definition 



F=k 



l ...dq n , (58) 



