20 Mr Sankey on the action of Caloric 



To apply this to a particular case, let the vessel be tetra- 

 hedral, its base being a square ; also let us suppose the mi- 

 nute particles of the fluid to be arranged directly one above 

 another. It is obvious that the number of atoms in each stra- 

 tum will be equal to the square of the number in each row of 

 that stratum ; so that, calling the number in each row «/, 

 X = If. Therefore, y^ s ■=. the entire number of atoms in the 

 fluid, which being a constant quantity, its differential ^sy^y 

 -f ?/- d 5 = o ; therefore 2 * d «/ = — y d s. But we have seen 

 above that hi — h or d A = ^ ( jo -f- f ) — j (ja + ^) = d ^ ( p + ^) 

 4-dj. d^-f-d^5. substituting therefore for d <9 its value — 



d /fc = ^— (04. t\ ^— d ^ 4- d ^ 5. But s 



y y ^^^ ^ y ^ 



h .1, 2dy h 2dy h , ^ dt h ^, 



= — — *.dh = ^ T~T, d t H —, . There- 



« dh 2dw Sdy.d^ dt ., „. 



fore, -, - =r ^ T^—r-T;. 4- — r". • Now, callmg any 



' h y y(^jiJ^t)^ pJ^t ^ ^ 



side of the square base of the vessel a^azny (p-j-^) — t -^^v, 

 or, — ^ -I- r; being indefinitely small, a = 2/ (p 4. » hence a 

 and p being constant, dy (p-f-^) -|- d^z/— o. Therefore, 



substitutmff for d y its value 7 ; -j- = -, + , . ,. a . 



^^ -^ pj^t'h- p-\.t ^ (p+t)"^ 



or, 7 — — ^ beinff indefinitely small, -7- — ; therefore, 



(p+t)2 & -> ' h p-\-t 



hyp log. /i = 3 hyp. log. jo + # + cor. When, however, ^ = 0, 



^v 7 7>^ X entire luimber of atoms in the fluid , 



then h — - — ^ , and p 4- 



t ^ p. Therefore, calling entire number of atoms w, cor, = 



n^ n 

 hyp. log. ' — 2 3 hyp. log. p ; therefore hyp. log. A z= 3 hyp. 



log. p -\-t + hyp. log. ^ ~ 3 hyp. log. p. 



If now we suppose equal quantities of caloric to be added 

 to the same fluid at different degrees of temperature, and to 

 exert equal energies in expelling the same number of particles 

 from each stratum, it is clear that the manifestation of these 

 energies, as estimated by the increased height of the fluid, 

 would be greater for the higher than for the lower tempera- 

 ture. We are not, however, authorized to assume that these 

 equal additions of caloric will exert equal energies. For whilst. 



