434 Mr. C. H. Lees on the Law of Cooling, and its bearing 



the form of (2 ; ) allows its variation to be taken into account 

 without materially affecting the integration. Writing 



c==c (l + cV), 



where c is the value of c at the temperature of the air in the 



experiments (about 17° or 18° C), and d is some constant, 



generally less than '001, (2 X ) becomes 



o v 



mc {l + dv) ^— = —sh .v\ 



ot 



Another small correction has to be applied for the change of 



temperature of the water-jacket, which up to the present has 



been assumed constant. Writing it now =V, where V is a 



function of t such that its value at the end of the experiment 



= 0, we have 



mc (l + dv) ^=-sh(v-Y) n . 

 ot 



From the Tables w r hich follow, it will be seen that V/v is 

 generally less than T ^ -; so that if, in the left-hand side of the 

 above equation, 1 + d(v — V) be substituted for 1 + dv, the error 

 introduced is generally less than yoooff* Also from these Tables 



dY . 



it is seen that, for an interval of twenty minutes, -rr— is with 



dv . ^ 



close approximation = —g ^— , where q is generally less than 



ot 



yJ-0. Hence, for an interval of twenty minutes, we have as a 



very close approximation, 



dv BV — — dv 



bt dt~ 1 ^ <J dt } 



or 



dv 1 dv-V 



dt 1 + q dt ' 

 and the equation of cooling corrected for all known variations 

 becomes 



— = — =e- — = — sliiv— V) n : 



1 + q dt : 



or 



|0_V)-*-r c '( v -yy-"\dv+ g7 < 1 + g) t = constant; 



J 7/1 C 



or 



(c _ V )-^fl- p 1 e'(v-Y)) = Shil + fejj t + C, (3) 



v J \ l—n v J mc Q ' 



Tables follow from which it can be seen how this equation 

 agrees with experiment. 



