4 M. L. A. Colding on the Universal Powers of 



the preceding remarks this will not be difficult, as we have seen 

 that the various kinds of energy are in reality not different, but 

 that all of them may be considered as one — for example, as me- 

 chanical energy. 



As we consequently have to determine the general mathema- 

 tical expression for the mechanical energy among the material 

 particles, or, which is the same thing, to determine the mathe- 

 matical expression for the total integral of vis viva which has 

 been called forth among these particles by an originally existing 

 source of motion, the following well-known example may be 

 useful. 



If a quantity of water m be at rest at the height h above the 

 surface of the earth, and if h is small enough for us to suppose 

 the force of gravity at the height h to be equal to the force of 

 gravity g at the surface of the earth, then it is a truth admitted 

 by all, and completely proved, that the whole integral of motion 

 which may be produced and imparted (for example, to a water- 

 wheel, or to any other machine) through the force of gravity will 

 be expressed by 



Q = m . g . h ; 



which effect, however, we shall only be able more and more nearly 

 to approach, never to obtain entirely, on account of the impedi- 

 ments which always occur, such as resistance of air, resistance of 

 friction, &c. As m .g is the weight of the water, and h is the 

 height through which the water is allowed to fall, we perceive that 

 itm.g is expressed in pounds and h in feet, then the mechanical 

 energy, which in mechanics generally is called the quantity of 

 work, is to be expressed, in foot-pounds — that is say, in pounds 

 raised one foot. 



Further, it is well known and proved that, if we abstract from all 

 those resistances which in fact will occur, then we obtain exactly 

 the same quantity of work, whether the water moves vertically in 

 the direction of gravity or is forced to move along any inclined 

 plane or any curved line through the height h ; the consequence 

 of which is that the increment of quantity of work dQ which is 

 developed by the falling through each little part ds of the path s 

 is equal to m .g multiplied by ds, resolved in the direction of the 

 force ; that is, 



dQ=m.g . -=- .ds. 



But it may easily be perceived that this formula would hold true in 

 general, even if the accelerating force g were any variable quan- 

 tity^' and m any mass, as g' will always remain constant during 

 the element of time dt in which the element of path ds is de- 

 scribed. If, then, we put the accelerating force resolved in the 



