in producing ewpanaion of Fluids and Solids. 19' 



vessel «, and the diameter of a minute particle p ; then h = 

 s{p+t)--t + v. 



If now we designate the increased altitude h'; the increas- 

 ed interval between the strata If; and that between the lowest 

 strata and the bottom of the vessel v', also the number of 

 strata s'; then in like manner, as above, we find that h! = 

 sf{p + t)-^t' + t;, consequently ^' — ^ = / {p + f) — ^' + t/ 



— 5 (jt? + + ^ — ^- ^"t w^ ^'^^^ seen above that the ad- 

 ditional number of strata, or / — s z=. ; substituting there- 

 fore for sf its value , h' — h = {xlf — xt -\- zt -^ zp) 



— tf J^ t ^ v' — V, or, because If — t and v' - — v may be sup- 

 posed very nearly equal, h' — h— {xlf — xt -\- zt -\- zp). 



But we have seen above that h=:S {p -{■ t) — t -^ v\ there- 

 fore s = — — , or, t — V being indefinitely small in re- 



spect of hfSz= , therefore, substituting this value for s, 



h' — A = X -^^ 4- . If now we consider h' — h as 



X — z p +i X — z 



indefinitely small, it will become the fluxion or differential of 

 the altitude or equal to d ^ ; also the number of atoms driven 

 off from each stratum will become the differential of the num- 

 ber in each stratum, or jsf = d or, and If — t will be the diffe- 

 rential of the distance between the strata or equal to d ^ /. 



d ^ =: , X — r-. -| T— » or d .r being indeiinitely small 



in respect of x, putting x for x — da:, and dividing by /*, 



-T- = . . H , therefore hyperbolic log. h = hyp. log. 



p •{- 1 -{■ hyp. log. X -{- cor. 



But further we may observe, that, as the section of the ves- 

 sel parallel to the base is constant, as also the size and form 

 of the particles themselves, consequently, if we suppose these 

 particles always to preserve their relative positions, which in- 

 deed appears necessary to the homogeneity of the fluid, the 

 number of particles in each stratum must depend on the dis- 

 tance between the particles, or x must be a function of t. 



