4 i2 SCIENCE PROGRESS 



Considering, first of all, the system of equations (9) we 



dx 

 see directly that the velocities -=- vanish with the co-ordinates x. 



Again, by differentiating (9) repeatedly, it is seen that 

 the second and all the higher derivatives also vanish together 

 with the first (and with the co-ordinates themselves). 



This vanishing of the accelerations with the velocities is, 

 of course, typical of an " inertia-free " or " completely damped " 

 system. 1 



On the other hand, the solution (10) appears at first sight 

 very similar in form to that obtained in the treatment of 

 small oscillations according to Lagrange. Such small oscil- 

 lations are not in general of the completely damped (inertia- 

 free) type ; they may not be damped at all, in which case 

 they represent a reversible process. 



There is, however, one essential difference which distin- 

 guishes the solution (10) here obtained from the solution of 

 the Lagrangian problem. In the latter, with a system requir- 

 ing for its definition n co-ordinates, there appear in the solu- 

 tion 2n arbitrary constants ; on the other hand, in the case 

 here considered, with n co-ordinates, there appear only n 

 such constants in the solution. This fact is closely related to 

 the fact that in a system possessing inertia it is, in general, 

 necessary, in order to specify the state of the system, to give 

 the values, not only of the co-ordinates, but of their first 

 derivatives (velocities) also. 



On the other hand, in an inertia-free or completely damped 

 system, such as we have here been considering, after the co- 

 ordinates are fixed, the velocities are no longer arbitrary, but 

 are fully determined. This fact finds expression ab initio in 

 equation (2) above, in which it is understood that the para- 

 meters P, Q do not include initial values of the first deriva- 

 tions of the X's. 



Equilibrium defined by Minimum Condition. — In mechanics 

 and thermodynamics the condition for equilibrium commonly 

 takes the form that certain functions (potentials) of the vari- 

 ables defining the state of the system assume a minimum 

 value. 



As already remarked, the phraseology thus borrowed from 

 mechanics and thermodynamics has been applied to bio- 

 logical systems, though no attempt has been made to give 

 such expressions the analytical form without which they 

 seem valueless or even meaningless. 



As our reflections here have been conducted along ana- 

 lytical lines, the question arises whether they enable us to 



1 See Buckingham, Thermodynamics, 1900, p. 33. 



