Principles of Dynamics. 179 



the sun and one star*. There are thus obtained _ — --- 



Distances, each of which bears the relation A to one of 

 the measurements. Now a long past experience, to which 

 there has been no exception, has revealed that whenever we 



obtain — - Distances, r ma . . . , by measurement on n 



particles at the same time, these distances are always roots 



o£ k - —{Sn — o) equations of the form f\(r m „ ...)=0, 



where the forms of the functions are the same for all sets of 

 measurements; so that, if ))» — 5 of the Distances are known, 

 the others can be found. This is the proposition which wo 

 mean to assert when we say that physical space has three 

 dimensions : if the number of such equations were 



— ^-— — '- — Qsn — 2N 4- 1), wc should say physical space had X 



dimensions. 



Accordingly we proceed to find 3vi — 5 independent varia- 

 bles, suitable functions of which we propose to introduce 

 Into our equations in place of the Distances. In practice, 

 for sake of symmetry, we find 3>i independent variables 

 (&m, y m i : »? ••• ) from on equations of the type 



O,,— .t^ + Om— y n f + {:,n — : n ) 2 =r 2 mn , . . (5) 

 and leave ourselves at liberty to introduce during some 

 subsequent stage of the proceedings five additional relations 

 between the quantities. 



The quantity V which is termed the "velocity of the solar 

 system in space " is defined by the following relations (the 

 suffix 1 in (5) above denotes that one of the particles 

 concerned in r lm is the sun): — 



V~r/+;y + ^ 2 J (6) 



m Jr*d , i 



a<3fc-*)|- • • • ( 7 > 



* I do not mean to assert that the distance of the sun from a star, or 

 the change of that distance, can be measured in the same way as the 

 distance between two particles in the laboratory. The distance between 

 two stars does not bear the relation A to any single measurement, but 

 it is, by definition, a known function of Distances which bear the relation 

 A to distances, the results of physical measurement; that is all that is 

 implied in the argument. 



N2 



V 



_ V 



