ON THE CONDUCTION OF HEAT. 15 



pothesis III. the form of the result in this case is « + — v'^ = a — fi z, whilst 

 the approximate form on hypothesis IV. is 



V Z)- = a — /3 Z. 



We may add, that if we assume k= aa^, the correct equation for hypothesis 

 III. will be a a rf j; = c, which gives 



— a =: C X + c'. 



log a 



When the time enters into the expression, no analogy exists between the 

 formulae, deduced on the two hypotheses. 



If we were in possession of a number of experiments adapted for the purpose, 

 it would be desirable that the constants in formula (12) and (13) were more 

 fully expressed. When such experiments shall have been made, a very little 

 trouble will suffice to effect this. Should it appear, moreover, that experi- 

 ments bearing on formulae different from that which we have given can be 

 more easily performed, the corresponding expressions may be readily deduced 

 from the equations given above. 



II. We proceed now to the second division of our report. We have to 

 inquire to what extent theory has been tested by experiment. In prosecuting 

 this inquiry we shall be led to examine for ourselves such of the formulae as 

 appear to be parallel to the experiments we possess. We hope by so doing 

 to point out the facility with which experiment is made available, as well as 

 practically to exhibit the neglect or want of attention which this department 

 of science has suffered. 



We turn first to the experiments described in M. Biot's ' Traite de Phy- 

 sique' (1816), tom. iv. Those which follow were made with the design of 

 testing the formula for the permanent state of heat at any point of a very 

 long bar heated at one end. The author himself compares them with the 

 formula, which, as we have said before, was discovered by him, but never 

 completely demonstrated. 



(1 .) The first experiment was made with a bar of iron, plunged at one extre- 

 mity into a bowl of mercury at 102§° cent. : along the bar were ranged eight 

 thermometers at various distances from each other. The observations were 

 made after the state of the temperature, as indicated by the thermometers, 

 had arrived at permanency ; and the length of the bar was so great that the 

 temperature of its free extremity did not sensibly differ from that of the air. 

 The following table contains the i-esults of this experiment reduced to the 

 centigrade scale. The distances are expressed in decimetres, and the num- 

 bers in the third column express the permanent state of the excess of tem- 

 perature of the corresponding points of the bar above that of the surrounding 

 air. This latter temperature was 1 6^°. 

 No. of therm. Dist. from Mercury. Excess of temp, above that of the air. 



86-25 



1 2-115 29-375 



2 3-115 17-5 



3 4-009 11-25 



4 4-970 7-1875 



5 5-902 4-6875 



6 7-777 2-1875 



7 9-671 1-25 



8 11-556 insensible 



