THE LAWS OF CONDUCTION OF HEAT IN BARS. 91 



79. A specimen of the Curves of Cooling is given in Plate IV. The subsidiary 

 curves in the same plate show different sections of the main curve projected on 

 different scales (as in the case of the Statical Curves, Art. 60), for convenience of 

 interpolation. The main curve corresponds to Case I. The dotted line adjacent 

 to the main curve in the Plate shows the modification of the law of cooling 

 introduced by covering the bar with paper, as in Case II. The results of the 

 whole are shown in the preceding Table. The origin of the abscissae (the times) 

 is of course wholly arbitrary in each case. 



80. The continuity of the curves thus obtained was in general satisfactory, 

 though in one or two instances it seemed desirable to project part of two curves 

 as distinct, as in Case I. 



81. It is of little use, however, to possess merely a knowledge of the free tem- 

 perature of the bar in terms of the time. The valuable information which we 

 require in the deduction of conductivity is the " rate of cooling," or the pro- 

 portional momentary loss of heat corresponding to a given excess of tempera- 



sjy 



ture. This is expressed mathematically by -j, and might be directly obtained by 



discovering the equation to the primary curve of cooling, and then differen- 

 tiating it. 



82. There is, however, not less difficulty in finding a formula of interpolation to 

 represent the curve of Cooling throughout its extent, than we have already found 

 in the case of the curve of Statical Temperature, and it would evidently require the 

 introduction of as many constants. I therefore preferred, in the first instance, 

 (seeing that from the multiplied observations of cooling, the ordinates of this curve 

 are more perfectly known than in the other case), to subdivide it into elementary 

 arcs, and treating each of these as a portion of a logarithmic curve (to which it 



approximates), to find the value of ^-, or the " rate of cooling," corresponding to 



successive values of %* and by projecting these in curves to study their inflections 

 in detail in each of the three forms of experiment already often referred to. 



83. The three upper figures of Plate V. represent the " rates of cooling" of each 

 bar in terms of its temperature-excess. From the study of these the peculiarities 

 of the law of cooling above adverted to will become evident, and the harmony 

 of the three cases is exhibited to the eye. 



84. First, For very small excesses of temperature, the rate of cooling is com- 

 paratively slow, but increases much more rapidly than the temperature. To illus- 



dv log v — log v v' + v 

 f By the formula — = 2-3026 — -, x — - — , where v and v' are the excesses of tempera- 



Civ h — Lt 



v' + V . 

 ture corresponding to the times t and t' . — — is the mean ordinate to which the result corresponds. 



The logarithms are tabular. 



