Principle in Thermodynamics. 433 



Supposing now that the last term on the right-hand side (that 

 is, the sum of the mean values of the variations resulting from 

 the change of form of the force-function) is =0, there remains 



Substituting 



ST = mv8v 

 and 



(tfx s , d*y s , d*z 5 \ dv . 



where Ss signifies the variation in the length of the path, we 

 obtain 



— m\ 



Now 



SSE=2 if' ro(t>&- —&)&. 



P^S ^ ft 55 f W,l, 



where the indices and 1 in the integrated portions denote the 

 initial and final values of the quantities. But as the path is 

 closed and the motion periodic, 



Ss 1 — Ss and v 1 = v . 

 Accordingly 



2SE = 2 1 Cm (vBv + v ^) dt. 



Taking into consideration that 



vdt — ds, 

 by a slight simplification we get 



^m = l}S l mh{fdt), 



If we bring the constant mass under the variation- symool and 

 reverse the succession of the integration and variation, we have 



1 Jo 

 or, designating the mean value by a horizontal stroke, 



i 

 Now this is the equation which we wished to demonstrate. 

 Phil Mag. S. 4. Vol. 46. No. 308. Dec. 1873. 2 G 



