S52 Ml*. W. J. M. llanklne on the Gons^miion of jEnei'pt/, 



tnotion, pi'oposecl a pair of analogous terms in tlie case of me* 

 chanical energy, viz. "virtual vis viva" and ''actual vis viva." 



6. Tlie principle of the conservation of energy may be thus 

 stated : — In any system of bodies, the sum of the potential and 

 actual energies of the bodies is never altered by their mutual 

 actions. 



AVhen applied to plisenomena purely mechanical, this is neither 

 more nor less than the long-known law of the conservation of 

 vis viva. The later discoveries respecting its applications have 

 reference to those cases in which the law of the conservation of 

 vis viva fails, and in which the increase or diminution of energy 

 in the mechanical forms is compensated for by the diminution 

 or increase of energy in other forms ; such as energy of heat, 

 which is the product of weight x temperature x specific heat 

 X Joule's equivalent ; energy of electric current, which is pro- 

 portional to electromotive force x quantity of current, or, other- 

 wise expressed, to (quantity of current)^ x resistance of cir- 

 cuit, &c. 



7. In the case of gravitation, the quantity which varies in- 

 versely as the square of the distance between a pair of bodies, is 

 their tendency to approach each other, to which no law of con- 

 servation applies. 



8. To find their energy, being the quantity which is conserved, 

 the following processes are to be gone through. 



To fix the ideas, let the bodies be spherical ; let a denote the 



sum of their radii, and m, m' their respective masses : let their 



tendency to approach each other, when their surfaces are in con- 



xmifnl 

 tact and their centres at the distance a apart, be denoted by — ^ . 



Then, when their centres are at any other distance apart, r, their 

 tendency to approach each other is 

 ._ Ymm' 



At any given instant, let r-^ be the distance between the cen- 

 tres of the bodies, and let their velocities, referred to their 

 common centre of gravity as a fixed point, be v, v'. (It is well 

 known that those velocities are contrary in direction, and in- 

 versely as the masses of the bodies.) Then 



I. To find the potential energy, construct a curve whose 

 abscissse are values of the distance r, and its ordinates values of 

 the attraction/, and take the area of that curve between the 

 ordinates whose abscissce are a and r^, that is to say, in symbols, 

 the potential energy 



:J>=F»..'(i-i). 



