Principles of Dynamics, 169 



seem that every possible and relevant observation on the 

 subject had been offered long ago. But so long as complete 

 agreement is not attained the discussion cannot he closed. 

 It is not long since Mr. Bertrand Russell, in his ' Principles 

 of Mathematics/ denounced Prof. Mach's conclusions, and 

 asserted that after all Newton's statement of the case was 

 superior to all the attempts that had been made to displace 

 it ; and, though probably all students of physics would agree 

 as to the main lines on which the principles of dynamics 

 should be stated, they display in approaching the questions 

 of Absolute and Relative Motion a hesitancy and uncertainty 

 which suggests that their ideas are after all not quite so clear 

 as they pretend. 



§ 2. There are two distinct problems, which are often 

 confused, which require solution. The first is " In what 

 manner are the fundamental propositions from which the 

 conclusions of theoretical dynamics are deduced to be stated ; 

 what are the conceptions employed in those propositions, and 

 what are the relations stated between them '? The second 

 is " In what manner are these propositions applied to experi- 

 ment ; how do we measure the magnitudes associated Avith 

 the fundamental conceptions ? " I believe that the difficulty 

 which has been found in answering these questions arises 

 from some confusion as to the relation between mathematics 

 and physics, and accordingly 1 propose to open the discussion 

 with some remarks on this relation. 



§ 3. The usefulness of mathematics to the student of ex- 

 perimental science arises from the following fact : — It is 

 sometimes possible to form mathematical equations containing 

 a certain number of quantities which are all capable of being 

 regarded as variables such that, when for some or! the 

 variables are substituted numbers arising from measurement 

 and the equations solved, the resulting values o£ the other 

 variables represent the numbers resulting from certain other 

 measurements. ] do not propose here to discuss how the 

 equations are to be formed, or in what cases they can be 

 formed : it will be sufficient for our purpose to note that 

 they are sometimes possible. 



§ 4. Let us take as an example, simpler in some respects 

 than that offered by mechanics, Van der AVaals's equation 

 for the state of a gas 



(?'+ < ;.)('- / o=i;t. 



There are six different quantities involved in this equation, 

 p, Vj T, a, h, R ; in order that the equation may be solved so 



