Principles of Dynamics. 173 



and state that the following relations hold between the old 

 and the new quantities : — 



72 i 



ffml+ ffma+ ffm* = Zjwd/m) } ' ' ' ' W 



70 

 /ij»l+ /i»»2+ »*«mi = 'ffi\ s m) 



and 



(.n„-^) 2 + (.y»-.'/») 2 + (-,»--«) 2 = '- (2) 



We now state the fundamental propositions of theoretical 

 dynamics in a series of equations, of which the following- 

 are typical : — 



wim/ ro „ + m»/n m ==0, and so fork's and h's, . (3 J 



/TOM '''m ,? '» J^WIM Jim ]h 



( Jmn l/m } J>i ''w/i -m ■-« 



« 



These equations are the answer to the first of the questions 

 of § 2 ; they represent the fundamental propositions or! 

 theoretical dynamics. Before proceeding to discuss the 

 second, and more important question, a few remarks as to 

 the nature of the quantities involved will he in place. 



Only the quantities r mn and t bear relation A to the 

 measures of physical quantities ; correlated with these Relative 

 Distances and Times there are relative distances and times. 

 The other quantities are not so correlated. The quantities 

 .t'm, t/m, Zm are termed the Absolute Coordinates of the par- 

 ticles, the quantities ■ {x m , //,„, z m ) their Absolute Accele- 

 rations, the quantities f mn their Partial Accelerations. These 

 quantities can only be defined by means of the equations of 

 dynamics ; so far as we have seen at present there are no 

 absolute coordinates or absolute accelerations to which they 

 bear the relation A*. 



§ 9. In the problems which are usually discussed by 

 theoretical dynamics a knowledge of the values of some of 

 the Masses and Partial Accelerations is assumed ; the others 

 are eliminated by the introduction of certain relations between 



* It is to be observed that there has been no necessity to introduce 

 Absolute Times, relaxed to the Relative Time t as the Absolute Coordinates 

 are related to the Relative Distances, because t enters into the equations 

 only in the differential coefficients ; we are only concerned with 

 differences of t. 



