INTO THE LAWS OF THE CONDUCTION OF HEAT IN BARS, ETC. 143 



The Reductions. 



27. The fundamental observations are those on the permanent temperatures of 

 a long iron bar heated at one end, and measured by thermometers placed at fixed 

 intervals along the bar (Art. 16, &c.). These were projected in a curve (which 

 approximates to, and has usually been treated as a logarithmic curve), of which 

 the length of the bar is the axis, and the temperatures (or rather excesses of tem- 

 perature above the surrounding space) the ordinates. This curve of statical tem- 

 perature was verified in all its parts as follows : — The first projection was made 

 from one of the completest and most satisfactory of the sets of observations, in- 

 cluding the temperature of ten or eleven points along the bar. Other sets of 

 observations were recorded, in which the fundamental temperatures differed more 

 or less from these, the source of heat being either more or less intense. But each 

 group of observations, while it may be treated separately, may also be regarded 

 as belonging to distinct parts of one common curve (so long as the bar and its 

 surface are unchanged). We are thus enabled to interpolate temperatures not 

 corresponding to observed points in the primary experimental curve, and to verify 

 them indefinitely. This process of interpolating independent groups of observa- 

 tions is peculiarly suited to the graphical method of reduction, and has been 

 pursued with entire success. 



28. Tangents were drawn and subtangents measured, and also calculated 



from ordinates, for different points of the statical curve. The table of — (where 



V is the excess of temperature of any point of the bar at a distance x from the 

 origin at the heated end) is thus formed (see Art. 6). The values being again 

 projected in terms of v, are graphically equalised. 



29. Next, the dynamical experiment of the cooling of the short bar (Art. 24) 

 is treated by projection; the temperatures being laid out in a curve in terms of 



(dv\ 

 j\, in terms of 



V, is calculated, and also projected and equalised. 



30. These last numbers are erected as ordinates upon the original line repre- 

 senting the axis of the long bar in the statical experiment, as in fig. 3, Art. 7. The 



perpendiculars are the values of yr, in terms of the ascertained measure of v at 



each point along the bar. The area of this last curve between the ordinate cor- 

 responding to X and infinity, represents the total loss of heat from the surface of 

 the bar on the cooler side of x, and consequently the whole flux of heat across the 

 section at x, since that flux is the quantity of heat necessary to compensate the whole 

 loss of heat from the surface of the har beyond x, the bar being, by hypothesis, in 

 a permanent condition. The determination of this area by the quadrature of the 

 successive parts is not difficult, as this curve also approximates to a logarithmic, 

 whence the area between successive ordinates, and also that adjoining the 

 asymptote, can be calculated with sufficient exactness. 



