526 Transactions. — Chemistry and Physics. 



20. We have here got two clear examples of what least 

 squares leads to. In the first case, that of the curve, as we 

 shall afterwards see, the shape of the curve of deviations is 

 most strongly indicative of the need for the application of a 

 formula of four terms, if not more. We have drawn the 

 deviations according to least squares, which may be proved 

 to arrange the deviations (given simply the direction of the 

 axis of X) from a cubic phenomenon to which a parabola is 

 applied, and where the observations are at nearly equal in- 

 tervals of X, with a symmetry similar to that of the graph. 

 The deviations, it will be observed, run + ( — , 0, + , 0, — , 0, + ) • 

 By the graphic process we should arrange the parabola so 

 that they run +(0, +,0, — , 0) — so that, in fact, they bear a 

 close resemblance to the standard cubic of Graph II. It is 

 asserted that there is less likelihood of systematic deviations 

 so arranged being mistaken for fortuitous errors than is the 

 case with the least-squares arrangement. It may be again 

 mentioned that this example is not, in its general features, a 

 mere hypothetical case. 



21. In the second case, that of six data, we have got a 

 curve from our least squares which we have asserted to be 

 quite unjustifiable, and not to be compared with the result? 

 that one would get from a common-sense judgment of the 

 graph — not to be compared, that is, in avoiding rash assump- 

 tions as to the truth of the matter. 



22. Following our definition of least squares, we have 

 neglected to take any account of fortuitous probable error in 

 these examples, but its vital necessity in such cases will be 

 sufficiently obvious from what has been said in previous sec- 

 tions. The effect of probable error in the graph is to obscure 

 the true points or line of the true curve of the phenomenon. 

 When this occurs to such an extent as to hide any system 

 there may be in the deviations, then, provided we are quite 

 sure that our formula is substantially accurate compared with 

 the scale of the probable errors, we might reasonably employ 

 least squares to systematize our computations. This is a 

 matter which is dependent upon circumstances, and more on 

 judgment, and we believe that the employment of the latter 

 will be found to be very largely dependent upon whether the 

 treatment of empirical formulae is taken as a mere extension 

 of the beautiful applications of the theory of probability to 

 astronomy and surveying or as a most important branch of 

 the graphical calculus. This part of the subject is too com- 

 plicated to treat of here except by suggestion, but we may 

 refer to the example in Dr. F. Kohlrausch's work (see sec- 

 tion 38), where a case of this complicated kind is given as if 

 it were a simple and logical application of least squares ; and 

 where, moreover, the data are deliberately subjected to extra- 



