12 Mr. Ivory on the Laxi:s of the Condemation and Dilatation 



whatever be the position of the small cylinder of air, and that 

 dz alone varies in different places of the tube, and at different 

 times. It follows therefore that x is independent on the time 

 t, and s is a function of x and t. It is to be observed too 

 that the air is supposed to undergo very small condensations 

 and rarefactions in proportion to its original bulk in the state 

 of equilibrium ; that is, dz must be considered as very small 

 when compared to d x. Let g' denote the density of the air 

 in equilibrio, and q the variable density of the agitated cylin- 

 der ; then, the masses of the two cylinders being the same, 

 their densities will be reciprocally as the volumes : therefore 



t (I I d z 



f ' dx ]-d% dx ^ 



the powers of the small fraction — ^ being rejected. This 



equation, it may be remarked, implies the continuity of the 

 fluid, since the cylinder in motion has always the same n)ass. 

 Let P' denote the elastic force of the air in equilibrio, and P 

 the like force of the agitated cylinder; then, if we adopt the 



law of Boyle and Mariotte, we shall have -^ = -—- : and this 



equation would lead us to the result obtained by Newton. 

 But if, according to the observation of Laplace, we reason 

 more agreeably to what actually takes place in nature, and 

 suppose that the elastic force of the agitated cylinder is ex- 

 erted while it retains the whole of its absolute heat, the pre- 

 ceding formulae (D) will furnish this equation, 



P' \ i / \ dx ) 3 dx 



Take the fluxions making x only variable, and divide by the 

 equal quantities q{dx + d z) and §'dx; then 



dP __4 P' ddz 



({dx + dz} ~ 3 ' e' ' dxi ' 



Now, P is the elastic force of the air in the tube at the di- 

 stance X + z from the assumed point in the axis, and P + rfP 

 is the like force of the air at the distance x + z-\-dx + dz; 

 wherefore d P is the effective force urging the intervening cy- 

 linder towards the assumed point : and as the mass moved is 

 equal to g{d x + d z), the quotient is the acceleration of each 



particle, otherwise expressed by — ^ ; 



wherefore '''^~ _ J_ _^' '^'^^ 



dri 3 ■ f' ■ dx'' ' 



Let I be the length of the homogeneous atmosphere, that is, 

 of a eohimn of air equal in weight to P' and having the same 



uniform 



