J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 347 



We may convince ourselves also of the general correctness of 

 this rule in the following way ; at least we may do so in a 

 preliminary manner. If ju. unknown magnitudes are to be found, 

 ju. equations of condition independent of each other are in every 

 case requisite; and if no more than jt* such equations are given, 

 we have no remaining standard for the estimation of the pos- 

 sible error. We do not obtain this until we substitute in other 

 equations of condition the values found instead of the unknown 

 magnitudes, and compare the errors which result ; so that with 

 m observations treated in this manner there result m — jx 

 errors, which allow us to form a judgement as to the exactness. 

 Inasmuch as we do not regard the ju. detenninate equations 

 alone as the absolutely correct ones, and the deviations of all 

 other results from those drawn from the j«. chosen results as 

 errors, but as we give to all an equal share in determining the 

 unknown magnitudes, we are certainly nearer the truth ; but we 

 do not by this means get rid of the analytical necessity of always 

 applying to the determination of ju. unknown magnitudes, if not 

 l>. determinate equations, yet an equivalent to such /x equations 

 taken from all together. Consequently the functions of the re- 

 maining errors thus obtained wUl always refer, not to a number 

 of m errors, but to the number of m — jx errors, as has been 

 shown for jx = 1, and as will be shown in the sequel for any ju. 

 taken at pleasure. 



A view of the rules for the heretofore considered simplest case, 

 i, e. the case of equally good direct observations of an unknown 

 quantity, — may be facilitated by applying them to Benzen- 

 berg's latest and most exact experiments on the fall of bodies, 

 made in the Schlebuscher coal mines. The object of these ex- 

 periments was to demonstrate directly the rotation of the earth 

 round its axis, by showing that balls let fall from a consider- 

 able height without initial velocity, deviate, in falling to the 

 lower station, towards the east, more than a plumb-line sus- 

 pended from the same upper point. The experiments, although 

 divided into separate parts, were so made as all to have the 

 same value. As they are only used as an example, I leave 

 quite out of consideration the deviation (not agreeing with 

 theory) of single balls towards the north and south, which 

 moreover almost entirely disappeared in the mean of all the 

 experiments. I also take as valid only those experiments which 

 the observer himself selects, — Table, page 424, ' Versuche 



2 A 2 



