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HELMHOLTZ ON THE CONSERVATION OF FORCE. 119 



mobile which could not only impart motion to itself, but also to 

 exterior bodies. 



If we inquire after the mathematical expression of this prin- 

 ciple, we shall find it in the known law of the conservation of 

 vis viva. The quantity of work which is produced and consumed 

 may, as is known, be expressed by a w^eight m, which is raised 

 to a certain height h-, it is then mgh, where g represents the 

 force of gravity. To rise perpendicularly to the height h, 

 the body m requires the velocity v= \^2gh, and attains the same 



by falling through the same height. Hence we have -rrw^^mgh; 



and hence we can set the half of the product mv^, which is 

 known in mechanics under the name of the vis viva of the 

 body m, in the place of the quantity of work. For the sake of 

 better agreement with the customary manner of measuring the 



intensity of forces, I propose calling the quantity -mv'^ the 



quantity of vis viva, by which it is rendered identical with the 

 quantity of work. For the applications of the doctrine of vis viva 

 which have been hitherto made this alteration is of no import- 

 ance, but we shall derive much advantage from it in the following. 

 The principle of the conservation of vis viva, as is known, de- 

 clares that when any number whatever of material points are set 

 in motion, solely by such forces as they exert upon each other, 

 or as are directed against fixed centres, the total sum of the 

 vires vivce, at all times when the points occupy the same relative 

 position, is the same, whatever may have been their paths or 

 their velocities during the intervening times. Let us suppose 

 the vires vivce applied to raise the parts of the system or their 

 equivalent masses to a certain height, it follows from what has 

 just been shown, that the quantities of work, which are repre- 

 sented in a similar manner, must also be equal under the con- 

 ditions mentioned. This principle however is not applicable to 

 all possible kinds of forces ; in mechanics it is generally derived 

 from the principle of virtual velocities, and the latter can only 

 be proved in the case of material points endowed with attractive 

 or repulsive forces. We will now show that the principle of 

 the conservation of vis viva is alone valid where the forces in 

 action may be resolved into those of material points which act 



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