128 Mr. Meikle on the 



librio, and § the variable density of the agitated cylinder ; then, 

 the masses of the two cylinders being the same, their densities 

 will be reciprocally as the volumes : therefore 



P — c?a? ^ ^ ^ dz 

 p' dx+dz dx* 



the powers of the small fraction -- being rejected*. This equa- 

 tion, it may be remarked, implies the continuityof the fluid f, 

 since the cylinder in motion has always the same mass. 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 -r-; = — ,- : and this equation 



would lead us to the result obtained by Newton J. But if, 

 according to the observation of Laplace, we reason more 



densation, in place of being trifling, must be infinite. For, here the length 

 of the cylinder is dx + dz, which binomial is likewise used as the fluxion 

 oi z ; no matter how curious and undefined the notation, which Laplace, 

 however, avoids. But when the cylinder reaches its utmost distance from 

 the assumed point from which x -^ z\s reckoned, and is about to return 

 toward that point, its velocity = ; and, therefore, the fluxion of the 

 space = dx + dz = 0, and dx = — dz. Or, more properly, dx—dz = 0, 

 and dx' = dz. For in this case, the fluxion of the space, or the length of 

 the cylinder, is obviously the difference and not the sum of dic and dz^ 

 because dx is constant. Hence, also, at the turn of the motion, the length 

 of the cylinder is nothing, or its density is infinite ; a consequence, though 

 absurd, yet inseparable from the tacit hypothesis which makes the cylin- 

 der always move over a space equal to its own length, during.the constant 

 fluxion of time dr. It is therefore certain, that the length of the cylin- 

 der cannot consistently represent its velocity, or coincide with the fluxion 

 of the space, as our author so conveniently assumes it to do, without 

 offering the least reason for such illegitimate procedure. It is almost 

 needless to add that the same assumption involves various other incon- 

 sistencies, or to remark that the shattering of windows and crazy build- 

 ings, the shaking of houses at considerable distances, the occasional 

 deafening of persons, with many similar effects, could neither be pro- 

 duced by small vibrations, nor slight condensations ; though infinite ones 

 would be unnecessary. — H. M, 



* Since, as we have seen, dz sometimes equals dx, this fraction is 

 occasionally considerable, or even equal to unit ; and, therefore, its powers 

 cannot warrantably be rejected, either here, or again a little after in 

 taking the fluxions. — H. M. 



']' True, a continuity, but only in one direction through the tube | 

 whereas, in the open air, the continuity is in all directions. — H. M, 



% We shall afterwards see this to be a mistake. — H. M. 



