at Temperatures lower than 32° F. 423 



tides of ice are formed at these centres of congelation, a large 

 quantity of heat is given out by each particle, and as water is a 

 bad conductor of heat, this is principally communicated to the 

 adjacent particles of water, aud when immediately afterwards 

 these particles pass from the liquid to the solid state, this trans- 

 ition takes place at a temperature higher than the initial one. 

 For a similar reason the liquid particles adjacent to these last 

 pass into the solid state at a temperature still higher ; and thus 

 the process continues until, at last, all the ice formed and the 

 surrounding water are raised to the temperature of 32° F. 

 The process of course would terminate here if the vessel con- 

 taining the liquid were a perfect non-conductor of heat, but as 

 this condition is unattainable, the reduced value in water of the 

 vessel must be found, and this included in the given quantity 

 of water. 



As the consecutive portions of ice are formed at different 

 temperatures varying according to a complicated law, which de- 

 pends on the conductibility of water, its latent heat, and the 

 specific heat of ice, the exact determination of the quantity of 

 ice formed would at first sight appear to be of considerable dif- 

 ficulty; this difficulty, however, as I shall presently show, is 

 only apparent. 



Before proceeding to the consideration of the problem itself I 

 shall establish the following theorem : — 



If \ denote the latent heat of water at the temperature 

 32°— 1°, X the latent heat of water at 32°, and c l the mean 

 specific heat of ice at the temperature 3%°—t°, then 



X 1= :X— (1 — c l )t 1 . 

 To prove this, consider one pound weight of water at the tem- 

 perature 32°— 1° 9 -and suppose it (1) converted into ice at the 

 same temperature, then (2) raised as ice in temperature to 32°, 

 then (3) converted into water at 32°, and finally (4) cooled down 

 agaiu as water to 32°~^°, and consequently brought back to 

 its original state. Now since the quantity of heat latent and 

 sensible in the pound of water is the same at the commence- 

 ment and end of the process, it is evident that the sum of the 

 two quantities given out by it in the stages (1) and (4) must be 

 equal to the sum of those communicated to it in the stages (2) 

 and (3). This consideration at once gives the equation 



\ 1 -\-t 1 =\ + c 1 t v or \ 1 = X—(l — c 1 )t 1 . . . (1) 



It may be remarked that no assumption is here made about 

 the specific heat of ice being constant for different temperatures ; 

 cj x is in fact only a symbol for the entire quantity of heat neces- 

 sary to raise one pound weight of ice from 32°— /j° to 32°. 



This being established, I shall now proceed to consider the 



