174 Mr. N. Campbell on the 



the Absolute Coordinates. These values must, of course, be 

 chosen so as to satisfy equations f3) and (4); these equations 

 play no further part in the discussion, which consists in 

 finding the relations between the Absolute Coordinates and 

 the Time by means of equations (1), the number of which is 

 equal to the number of Absolute Coordinates. The Relative 

 Distances can then be expressed as functions of the Time by 

 means of the equations (2). But in the use of theoretical 

 mechanics for the purposes of experimental science the most 

 important question is that of which the answer is assumed in 

 dealing with such problems ; the object of the investigation 

 is to find out what values are to be assigned in any special 

 case to the Masses and the Partial Accelerations, and these 

 values, if they are to be determined at all, must be deter- 

 mined from the fundamental equations. For, as has been 

 pointed out, they cannot be determined directly from measure- 

 ment by means of our relation A, nor can they possibly be 

 determined by the solution of any other equation, since the 

 equations of dynamics are fundamental and logically prior 

 to any others involving the same concepts. We must decide 

 then whether it be possible, and, if so, by what means it is 

 possible, to determine the Masses and Partial Accelerations 

 from a knowledge of the relation between the Relative Dis- 

 tances and the Time, such as is deduced from the observed 

 relation between the relative distances and the time. 



It is clear immediately that a unique determination of 

 these quantities is not possible in general. For suppose 

 that we have made N observations on a system of n particles, 

 giving N sets of corresponding values of r mn and t. Then we 

 have 3N->i equations of the type (1), N(3rc — 5) of the type (2) 



(see § 12 below), — -n(n— 1) of the type (3), and Nii(h- 1) 



~ / a i \ 



of the type (4), in all ( r } n 2 + — 5 )N. On the other hand, 



the number of unknown quantities, which must be treated as 

 independent variables, is 3n 3 N + 72 — 1, that is 3N>i(n — 1) 

 Partial Accelerations, 3Nn Absolute Coordinates, and n— 1 

 Masses, since we are content with a knowledge of the ratios 

 of the Masses*. 



It is to be observed that the number of equations will be 

 greater than that of the independent variables for some 

 values of N and n, and will be less for others. 



* Really the number of equations ought to be somewhat greater than 

 is stated, for it is assumed implicitly that all the masses are positive 

 quantities. The estimation of the additional number of equations to be 

 added on this account, if it is possible at all, is very difficult and would 

 not affect the result of the argument. 



