CHAPTER II. 



APPLICATION OF THE PRINCIPLE OF CONSERVATION 



OF EXTENSION-IN-PHASE TO THE THEORY 



OF ERRORS. 



LET us now proceed to combine the principle which has been 

 demonstrated in the preceding chapter and which in its differ- 

 ent applications and regarded from different points of view 

 has been variously designated as the conservation of density- 

 in-phase, or of extension-in-phase, or of probability of phase, 

 with those approximate relations which are generally used in 

 the 'theory of errors.' 



We suppose that the differential equations of the motion of 

 a system are exactly known, but that the constants of the 

 integral equations are only approximately determined. It is 

 evident that the probability that the momenta and coordinates 

 at the time t' fall between the limits pj and pj + dp^ q^ and 

 q-L + dq^ etc., may be expressed by the formula 



e* d Pl ' . . . dqj, (48) 



where rf (the index of probability for the phase in question) is 

 a function of the coordinates and momenta and of the time. 



Let Qi, P^t etc. be the values of the coordinates and momenta 

 which give the maximum value to ?/, and let the general 

 value of rj be developed by Taylor's theorem according to 

 ascending powers and products of the differences p^ P/, 

 Q.I ~ Ci' Q te"> an( i let us suppose that we have a sufficient 

 approximation without going beyond terms of the second 

 degree in these differences. We may therefore set 



n' = c F', (49) 



where c is independent of the differences p^ P/, q{ /, 

 etc., and F 1 is a homogeneous quadratic function of these 



