Mr. W. J. M. Rankine on the Mechanical Action of Heat. 15 



molecular vortices, the amount of heat produced by an indefinitely 

 small compi'ession of one atom of a body in that state of perfect 

 fluidity which admits of the bounding surface of the^atom^being 

 treated as if it were spherical ; its radius being denoted by R, 

 and the radius of any internal spherical layer of the atmosphere 

 by multiplying R by a fraction m*. 



I shall denote by the ordinary symbol of differentiation d, such 

 changes as depend on the various positions of portions of the 

 atomic atmosphere relatively to each other, when changes of 

 volume and temperature are not taken into consideration ; while 

 by the symbol S of the calculus of variations, I shall represent 

 such changes as arise from the variations of volume and tempe- 

 rature. 



Let us consider the case of an indefinitely thin spherical layer 

 of the atomic atmosphere, whose distance from the nucleus is Rm, 

 its thickness Rrfw, its area 47rR^M^, and its density 



The weight, then, of this layer is 



47rR3 ^ Dm^-^It («, D, t) du. 



Its velocity of oscillation is v, and having, in virtue of that 

 velocity, a mean centrifugal force, as explained in the Introduc- 

 tion (Equation V.), equal to 



it is kept in equilibrio by an equal and opposite centripetal force, 

 arising from attraction and elastic pressui'e, which is consequently 

 represented by 



47rR2^|^D«Vr(r^,D,T)^« 



= 87rR2 ^ QDmi^ (m, D, t) du. 



Let the mean density of the atom now be increased by the 

 indefinitely small quantity 8D. Then the layer will approach 

 the nucleus through the distance — 8(Rm) = — MSR^R8^^, and 

 being acted upon through that distance by the centripetal force 

 already stated, the vis viva of oscillation will be increased by a 



* In a subsequent paper, read to the Royal Society of Edinburgh on the 

 15th of Ueecmber 1^(51, I have shown, that the same results, thoufjh by a 

 more complicated analysis, are obtained, su])posing the boundaries of the 

 atoms to be of any figure whatsoever. The su))position that they are sjiheri- 

 cal, therefore, is to be regarded as merely an expedient to simplify calcula- 

 tion, and not as an essential part of the theoi-y. (See Trans. Royal Soc. 

 Edinb. vol. XX. part.'l) 



