THOMSON AND TAITS NATURAL PHILOSOPHY. 783 



any particular values, and the only scientific way of dealing with them is to 

 ignore them. 



But this is not all. Since these variables do not appear in the expression 

 for the potential energy, there can be no force acting on them, and therefore 

 their momenta are, each of them, constant, and their velocities are functions 

 of the variables, but, since their own variables do not enter into the expressions, 

 we may consider them as functions of the other variables, or, as they are here 

 called, the retained co-ordinates, and of the constant momenta of the ignored 

 co-ordinates. 



From the velocities as thus expressed, together with the constant momenta, 

 we obtain the contribution of the ignored co-ordinates to the kinetic energy 

 of the system in terms of the retained co-ordinates and of the constant momenta 

 of the ignored co-ordinates. This part of the kinetic energy, being independent 

 of the velocities of the retained co-ordinates, is, as regards the retained co- 

 ordinates, strictly positional'", and may be considered for all experimental purposes 

 as if it were a term of the potential energy. The other part of the kinetic 

 energy is a homogeneous quadratic function of the velocities of the retained 

 co-ordinates. In the final equations of motion neither the ignored co-ordinates 

 nor their velocities appear, but everything is expressed in terms of the retained 

 co-ordinates and their velocities, the coefficients, however, being, in general, 

 functions of the constant momenta of the ignored co-ordinates. 



We may regard this investigation as a mathematical illustration of the 

 scientific principle that in the study of any complex object, we must fix our 

 attention on those elements of it which we are able to observe and to cause 

 to vary, and ignore those which we can neither observe nor cause to vary. 



In an ordinary belfry, each bell has one rope which comes down through 

 a hole in the floor to the bellringers' room. But suppose that each rope, in- 

 stead of acting on one bell, contributes to the motion of many pieces of 

 machinery, and that the motion of each piece is determined not by the motion 

 of one rope alone, but by that of several, and suppose, further, that all this 

 machinery is silent and utterly unknown to the men at the ropes, who can 

 only see as far as the holes in the floor above them. 



Supposing all this, what is the scientific duty of the men below? They 

 have full command of the ropes, but of nothing else. They can give each 

 rope any position and any velocity, and they can estimate its momentum by 

 * The division of forces into motional and positional is introduced at p. 370. 



