134 PEINCIPAL J. D. FORBES ON AN EXPERIMENTAL INQUIRY 



along the axis of the bar, the temperatures having assumed a permanent con- 

 dition. In conformity with this assumption, it is farther taken for granted that 

 the excesses of temperature of the bar above that of the air may be correctly 

 represented by a logarithmic curve, whose asjmptote is parallel to the length of 

 the bar. In order that this may be allowable, it is also necessary to assume that 

 the exterior loss hy radiation is at every point proportional to the excess of temper- 

 ature, which (especially when the convective action of the atmosphere is taken 

 into account) is known to be wide of the truth. To assume the logarithmic form 

 of the temperature curve, and to compel it to represent the observations, by sub- 

 jecting the latter to the Procrustean ordeal of the method of '-' minimum squares," 

 as was done by Biot, might lead, no doubt, to approximate numerical constants, 

 but could never serve to revise and rectify our knowledge of the laws of conduc- 

 tion.* Besides, the results of the statical form of the experiment cannot by 

 itself lead to any estimate of absolute conducting power. 



4. The plan which I devised in 1850, if not earlier, is plainly specified in the 

 following extracts from letters addressed by me to Professor Kelland in that 

 year, which I shall here transcribe. I may add, that about the same time I 

 transmitted a similar statement to Mr Airy, with a request that it might be pre- 

 served among the scientific correspondence of the Royal Observatory. These 

 communications were referred to in a provisional report to the British Associa- 

 tion, dated 19th June 1851.f 



5. The following is an exact transcript of the portions of the letter first 

 referred to above. 



To Professor Kelland. 



Phoesdo, hy Laurencekirk, 2Gth Sept. 1850. 



" I. I propose to find experimentally the effect of external cooling, from 

 all causes whatsoever, on the interior temperature of a uniform bar, heated 



* The fallacy of tlie method of minimum squares, as it is often applied to physical inquiries, is 

 well exemplified in this experiment, and may be seen almost at a glance by inspectino- Biot's com- 

 parative tables of calculated and observed temperatures, given in his Tralte de Fhysique, vol. iv.. 

 p. 666. The " probable error" of an observation of temperature at each and every point of the bar is 

 considered as the same ; whereas, when the excesses of temperature are only a few degrees, every ex- 

 perimenter knows that they may be determined with the nicest accuracy, whilst, when the excesses 

 rise to hundreds of degrees, the uncertainty is incomparably greater. It is quite evident, that Biot's 

 observations differ systematically from the logarithmic law. In referring in 1856 to these observa- 

 tions [Encyclopaedia Britannica, Sixth Dissertation, Art. (671)], I observed, that "instead of draw- 

 ing from them, as he does, an argument for the accuracy of the Newtonian law of cooling, the dimi- 

 nution of temperature along the bar is far more rapid at first, and less afterwards, than that law 

 indicates. In fact, the apparent agreement of the formula is owing to the use, in a case to which it 

 does not correctly apply, of that often misapplied rule of tlie doctrine of chances — the method of 

 least squares." See also Mr Airy's Theory of Errors of Observations (1861). 



I Report for 1851 (Ipswich). Sections, p. 7. 



