ON THE GENERAL THEORY OF THE STEAM ENGINE. 183 



II. PRINCIPLE OF LIVING FORCES. 



The quantity of power expended in the production or maintenance of any effect is, how- 

 ever, most simply determined by the general principle of living forces. This principle, which 

 is immediately deducible from that of virtual velocities, is applicable to every possible motion 

 of terrestrial bodies, and may be enumerated as follows : 



At any instant let each particle of the system be multiplied into the square of its velocity, 

 and the product will be the living force of the particle. The aggregate of all these products 

 will measure the total amount of impetus or living force of the system. 



Let each force CJ, acting at a particular point, be multiplied into the small distance dq tra- 

 versed by that point, in the direction of the line in which the force acts, during an elementary 

 instant of time dt, and the product will measure the expenditure of power during that instant. 

 In this measurement, the forces or pressures arising from the connexion of the parts will be 

 neglected, because no motion can at any instant be produced in the direction of such press- 

 ures, and they consequently can have no share in the generation or consumption of living 

 force ; but the resistances of friction, which are proportional to the pressures on the parts, 

 and act in the direction of the motion, must be regarded. In the cases where the point 

 moves in the direction of the force, the effect will be to increase the living force ; but in the 

 cases where the point moves in the contrary direction, the effect will be to diminish the 

 living force, and all resistances are of this latter description. 



Now, if from double the total expenditure towards the increase of the living force we 

 deduct double the total expenditure towards its diminution, each being so estimated, the 

 difference will always accurately express the variation that takes place, during the instant dt, 

 in the above-defined value of the living force of the system. 



The sum of the variations through successive instants will express the total variation that 

 takes place in a finite time. When the forces are variable, this summation must be effected 

 by tne integral calculus ; but when the forces are constant, it will only be necessary to mul- 

 tiply each into the whole distance traversed by its point of application in the direction of its 

 action, to get its effect in any given time or space. 



When the motion is permanent and the velocities recur at stated periods, the whole vari- 

 ation of living force through the extent of each period must evidently be zero ; and hence in 

 this case the expenditure of the moving powers must be precisely equal to that of the 

 resistances throughout an entire period. 



This principle, though exceedingly general, is of very easy application, and it will readily 

 be perceived how the preceding theories of the action of the crank, or indeed of any other 

 action, may be immediately deduced from it. 



It may be here worthy of remark, that in the case of steam employed at a given temperature, 

 the density being inversely as the pressure, the expenditure of the moving power of the 

 piston will be proportional to the quantity of steam consumed or the effective quantity of 

 water evaporated. 



