on certain Equations in the Analytical Theory of Heat. 433 



minutes of the curve of cooling, when the mean temperature- 

 excess of the bar was about 64° C, it is always 30 or 40 per 

 cent, greater than a determined from the last eight minutes 

 when the temperature-excess was 11° C. Hence a increases 

 with the temperature, and we are led to the assumption : — 

 f(v)=v(l + bv)*, which gives 



I-— — — . +at = constant, 

 Jv{l + bv) 



or 



— hb=Ae at , where A is an arbitrary constant. 

 v 



The constants of this equation, determined from observations 

 at times 0, 10, 20 minutes, are in one experiment 



a=-0277, 6 = -0047, A = '0187; 



and at 40, 50, 60 minutes, in the same experiment, 



a = -0221, 6= -0165, A=-0284. 



A similar variation was found in other cases, so that the 

 cooling is not well represented by making f(y) a quadratic 

 function of v. 



Ceasing to consider integral powers of v, we write 



f(v)=v(a + bv m ), 



where m is some + quantity. This gives as integral, 



v~ m + - =Ce amt . 

 a 



Solving this by trial we find m = '2 approx. and a = ; and 

 we deduce as probable form/(v) = v n , where n> 1. 

 The equation (2) therefore takes the form : — 



cm^ = -sh.v n ; (2') 



or the rate of loss of heat from the bar varies as the nth 

 power of the excess of temperature of bar above temperature 

 of aii*, supposed to remain constant, where n=l'2 approxi- 

 mately. 



So far the specific heat c has been considered constant ; but 



* Kundt and Warburg (Pogg. Ann. clvi.) make use of this to express 

 the cooling of a thermometer in a sphere concentric with its bulb. H. F. 

 Weber (Mon. Ber. d. Berlin Akad. 1880) considers some correction of this 

 form to be necessary in dealing with conductivities of bars. 



