532 Transactions. — Chemistry and Physics. 



additive functions.* It is thought that the statement in 

 Thomson and Tait's " Natural Philosophy " (ed. 1890, p. 454) 

 of an interpolation method, and the reference to " a patient, 

 application of what is known as the method of least squares " 

 in Professor Perry's "Calculus for Engineers" of 1897 

 (p. 18), form a sufficient ground for this conclusion. 



Conclusion. 



38. Those who may be inclined to question the necessity 

 of such remarks as have been made upon an admittedly 

 insufficient definition of least squares are recommended to 

 examine, in the light of the considerations that have been 

 advanced, the example of least squares put forth in Dr. F. 

 Kohlrausch's work, English translation (called " Physical 

 Measurements "), of 1894, from the German of 1892 (7th ed., 

 chap. 3), and also Professor Merriman's " Theory of Least 

 Squares" (1900 edition), with reference to Clairault's formula 

 (about page 126), and from page 130 to the end of the 

 Mississippi Problem. If, also, it is desired to observe how 

 even legitimate least squares may lead to error, an examina- 

 tion may be made of the warnings of Sir G. B. Airy in the 

 conclusion of his work on the " Theorv of Errors of Observa- 

 tion, &c." (pp. 112, 113). t The 1874 "edition of this work is 

 available in the Public Library. A paper by F. Galton, 

 F.E.S., in the "Proceedings of the Royal Society" of 1879, 

 page 365, also contains a significant warning that the funda- 

 mental principle of the arithmetic mean is not always reliable. 

 This should be considered in relation to the use of a curve 

 of dY/dX. in treating measures where X cannot conveniently 

 be adjusted to the desired datum point for every observa- 

 tion. 



39. We may also venture on the suggestion that, while 

 many writers have been quite wrong in calling the constants 

 of an empirical formula the "most probable ones," those 

 who have called them "the best" merely may have been 

 quite justified in making use of such an expression where it 

 has not been shown that analytical resources of greater power 

 are available, as has been the case with the Fourier series, 

 and it is hoped will be now seen to be the case with the 

 Taylor series and other linear additive formulae. Further, 

 the habit of referring to empirical formulas as "laws" may 

 have helped to give such formulae an importance which, com- 

 pared with the graph, they assuredly do not possess. 



* It should be mentioned, however, that Professor Callendar (Phil. 

 Trans., 1887, p. 161) uses the formula of the standard parabola in con- 

 nection with the reduction of platinum thermometry. 



f In this connection see section 4. 



