THE LAWS OF CONDUCTION OF HEAT IN BARS. 109 



128. Suggestions as to Reductions. — Were any one desirous of pursuing the 

 subject of the theory of conduction into its details, I should be disposed to 

 recommend the employment of Biot's formula of 5 constants (used to express 

 the elasticity of steam), instead of Roche's, containing 3 constants, which we 

 have here used, see Art. QQ. The method of calculation (which is necessarily 

 laborious), is given in Regnault's large treatise on the Theory of the Steam 

 Engine* For any merely practical purpose, however, this is not required. An 

 experimenter desiring to compare the conductivity and " percentage decrement" 

 of different metals, may reasonably confine his attention between the useful 

 limits of 20° and 120°, or at most 140° Centigrade. For that interval, Roche's 

 formula will suffice. And the chief use of the formula is to obtain readily and 



accurately the differential co-efficient j- (see Arts, 71 and 78), on the determina- 

 tion of which the value of the conductivity mainly depends. 



129. Though I would not recommend the attempt to proceed by graphical 

 methods alone, they are an invaluable help, and also serve as a check to the 

 calculations. Where these are not made throughout in duplicate, the use of 

 curves ensures the detection of any material error of the computer. The check 

 by taking first and second differences should also not be disregarded. The curves 

 of cooling may be treated in a similar way. 



130. I believe, however, that very fair results might be rapidly and approxi- 

 mately obtained by graphical methods alone. The curves of Statical Temperature 

 and of Cooling being first projected in the usual way, tangents might be drawn 

 mechanically for ordinates successively differing by 10°. The ordinate divided 



by the subtangent found would give the numerical values of -r- and j They 



would no doubt be somewhat irregular from the clumsiness of the graphical 

 process; but being projected in terms of x and v respectively, and equalizing 

 curves drawn through them, fair results would be obtained-! The " statical curve 

 of cooling" is then constructed without any calculation whatever ; and for evalu- 

 ating its area up to any limiting ordinate, it might be sufficient that the curvilinear 

 space it encloses should be defined on writing paper and cut out with scissors : 

 the successive portions being weighed, would represent the flux of heat in known 



* I ought perhaps to mention the formula which Professor Rankine has applied with success 

 to express the elasticity of steam at all temperatures (Edin. Phil. Journ. 1849, vol. xlvii. p. 28, and 

 Philos. Mag. 1854, vol. viii. p. 530). It is as follows: — 



log P = A - -- % 

 r r 



where P is the elasticity of vapour, and r the temperature reckoned from an absolute zero ( — 274° 

 cent). In applying the formula to the temperature of a bar, there can be no natural zero from 

 which the lengths are reckoned along the bar; and therefore the constants, instead of three in 

 number, may be reckoned as four ; putting v instead of P in the above formula, and, instead of r. 

 writing a; + D, D being some fourth constant. (See article 67.) 

 f This method was used by me in 1852. 



VOL. XXIV. PART I. 2 G 



