THEORY OF ERRORS. 21 



differences. The terms of the first degree vanish in virtue 

 of the maximum condition, which also requires that F' must 

 have a positive value except when all the differences men- 

 tioned vanish. If we set 



0=ef, (50) 



we may write for the probability that the phase lies within 

 the limits considered 



d Pl > . . . dqj. (51) 



C is evidently the maximum value of the coefficient of proba- 

 bility at the time considered. 



In regard to the degree of approximation represented by 

 these formulae, it is to be observed that we suppose, as is 

 usual in the 'theory of errors/ that the determination (ex- 

 plicit or implicit) of the constants of motion is of such 

 precision that the coefficient of probability e* or Ce~ F ' is 

 practically zero except for very small values of the differences 

 Pi P 1 / , q^ Ci'> e ^ c< For very small values of these 

 differences the approximation is evidently in general sufficient, 

 for larger values of these differences the value of Ce~ F ' will 

 be sensibly zero, as it should be, and in this sense the formula 

 will represent the facts. 



We shall suppose that the forces to which the system is 

 subject are functions of the coordinates either alone or with 

 the time. The principle of conservation of probability of 

 phase will therefore apply, which requires that at any other 

 time (t") the maximum value of the coefficient of probability 

 shall be the same as at the time t\ and that the phase 

 (Pi', Qi'-) etc.) which has this greatest probability-coefficient, 

 shall be that which corresponds to the phase (P/, -/, etc.), 

 i. e., which is calculated from the same values of the constants 

 of the integral equations of motion. 



We may therefore write for the probability that the phase 

 at the time t" falls within the limits p^ 1 and p : " + dp^ #/' 

 and #/' + cfy/', etc., 



" dpi" ...dqj', (52) 



