208 Dr. 0. J. Lodge on the Variation of the Thermal 



rature of any point of the rod over that of the enclosure. It 

 may be any thing between 0° and 150°; but it is not likely in 

 ordinary experiments to go above 200°. The terms of the 

 above series for the extreme case 6 = 200 are 



267 + 200 + 100 + 40 + 1, 



where only the last term, containing the fourth power of the 

 temperature, can be regarded as quite negligible. But for the 

 more likely case of = 100, the terms of the series are 



267 + 100 + 25 + 5+ ^, 



where the term containing the cube of 6 is not of much con- 

 sequence. If, however, it were wished not to go higher than 

 the second power, the term containing the cube need not be 

 neglected, but a mean value of it may be added to the coeffi- 

 cient of the second power of 6. Thus if © be the highest 

 temperature taken notice of (i. e. the temperature of the origin 

 in the case of the rod, the initial temperature in the case of 

 a cooling body), 



and this is the expression we shall use, writing it first in the 

 simpler form, 



»-(»{ BOT + » + 5,(l + -^)}. ... (7) 



Notice that 6 occurs in this expression as a factor, so that it 

 is really a cubic function of 6. 



It is singular how near the constant term in these brackets is to 

 the number 274. I suppose this is accidental ; but at first sight it 

 looked as if the rate of cooling for small excesses of temperature 

 were proportional to the product of absolute temperature and ex- 



cess, or as if the quotient — would be constant. On this hypo- 

 thesis, however, the constant a, twice the reciprocal of whose loga- 

 rithm is the number which happens to be nearly 274, would vary 

 with v the temperature of the enclosure, which is contrary to Du- 

 long's results. Indeed there seems no ground for the conjecture. 



Applying the correction for the neglected terms of the series, as 

 is done in (7), we may write the expansion (6) thus, writing a, in- 

 stead of log e a for shortness (a=*0076), 



^-l-=a0Jl+| a 0+^a 2 2 /'l+^ae^|. . . (7') 



13. It remains now to show^ from Tables of experimental 

 results, to what amount of accuracy 0, multiplied by a quadratic 



