134 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



phenomena, we have made a distinct advance in our knowledge of 

 the subject. A striking example is furnished by the kinetic 

 theory of gases, by means of which we are enabled to pass from 

 the bare statement that all gases press against their confining 

 vessels to the statement that this pressure is due to the impact of 

 the molecules of the gas against the walls of the vessel. 



69. Properties of Cyclic Systems. Reciprocal Rela- 

 tions. Since by the properties of the kinetic energy we have 

 three different kinds of quantities represented by partial deriva- 

 tives of one or the other of two functions, 



> v TJ dT , . . dT / \ T dTp , . - , 



applying the principle that a derivative by two variables is inde- 

 pendent of the order of the differentiations we obtain six reciprocal 

 theorems. We shall throughout suppose that there is no potential 

 energy. 



I a. In an adiabatic motion if an increase in one positional 

 coordinate q r causes an increase in the impressed force P s belong- 

 ing to another positional coordinate q s at a certain rate, then an 

 increase in the positional coordinate q s causes an increase in the 

 impressed force P r at the same rate. For 



dq r 



I 6. In an isocyclic motion we have the same property as 

 above. For 



(6) 



dq r dq r dq 8 dq s ' 



II a. If in any motion an increase of any cyclic momentum 

 p r , the positional coordinates being unchanged, causes an increase 

 in a cyclic velocity q s ' at a certain rate, then an increase in the 

 momentum p s , the positional coordinates being unchanged, causes 

 an increase in the velocity q r ' at the same rate. For 



dp r dp$p s dp s ' 



II 6. If in any motion an increase in any cyclic velocity q r ', 

 the positional coordinates being unchanged, causes an increase in 

 a cyclic momentum p s , then an increase in the velocity q s ' causes 



